M1- Multiplying matrices


Example 1

M1- Multiplying matrices (ver. 1)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -1 & -1 & 2 & 1 \\ -1 & -2 & 2 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & -1 & 0 & 3 \\ 1 & 0 & -2 & -1 \\ 0 & 0 & 1 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -5 & -4 \\ 0 & 1 & 1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -5 & -4 \\ 0 & 1 & 1 \end{array}\right] \left[\begin{array}{rrrr} 1 & -1 & 0 & 3 \\ 1 & 0 & -2 & -1 \\ 0 & 0 & 1 & 1 \end{array}\right] = \left[\begin{array}{rrrr} -4 & -1 & 6 & 4 \\ 1 & 0 & -1 & 0 \end{array}\right]\]


Example 2

M1- Multiplying matrices (ver. 2)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 2 & 2 & 0 \\ -2 & -3 & -4 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 5 & -3 \\ 2 & -1 \\ 4 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 1 & -1 & 4 \\ 0 & 1 & -1 & 1 \\ 1 & -2 & 3 & 3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 5 & -3 \\ 2 & -1 \\ 4 & -6 \end{array}\right] \left[\begin{array}{rrrr} 1 & 2 & 2 & 0 \\ -2 & -3 & -4 & -1 \end{array}\right] = \left[\begin{array}{rrrr} 11 & 19 & 22 & 3 \\ 4 & 7 & 8 & 1 \\ 16 & 26 & 32 & 6 \end{array}\right]\]


Example 3

M1- Multiplying matrices (ver. 3)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -5 & -3 & 1 \\ -3 & -2 & 0 \\ 0 & -2 & -5 \\ -2 & -2 & -2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 1 & 4 & 6 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 2 & -2 \\ 0 & 1 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 1 & 4 & 6 & 3 \end{array}\right] \left[\begin{array}{rrr} -5 & -3 & 1 \\ -3 & -2 & 0 \\ 0 & -2 & -5 \\ -2 & -2 & -2 \end{array}\right] = \left[\begin{array}{rrr} 0 & 2 & 5 \\ -23 & -29 & -35 \end{array}\right]\]


Example 4

M1- Multiplying matrices (ver. 4)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} -2 & 2 \\ 1 & 2 \\ -3 & 1 \\ -1 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -3 & -6 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \\ 5 & -5 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -3 & 2 & -4 \\ 4 & -3 & 6 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -2 & 2 \\ 1 & 2 \\ -3 & 1 \\ -1 & -6 \end{array}\right] \left[\begin{array}{rrr} -3 & 2 & -4 \\ 4 & -3 & 6 \end{array}\right] = \left[\begin{array}{rrr} 14 & -10 & 20 \\ 5 & -4 & 8 \\ 13 & -9 & 18 \\ -21 & 16 & -32 \end{array}\right]\]


Example 5

M1- Multiplying matrices (ver. 5)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} -1 & -5 \\ 1 & 4 \\ 3 & 6 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 0 \\ 2 & 1 \\ 1 & 0 \\ 5 & -4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -5 & 1 & 6 \\ 2 & -1 & -4 \\ 0 & -2 & 1 \\ -3 & 1 & 4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} -5 & 1 & 6 \\ 2 & -1 & -4 \\ 0 & -2 & 1 \\ -3 & 1 & 4 \end{array}\right] \left[\begin{array}{rr} -1 & -5 \\ 1 & 4 \\ 3 & 6 \end{array}\right] = \left[\begin{array}{rr} 24 & 65 \\ -15 & -38 \\ 1 & -2 \\ 16 & 43 \end{array}\right]\]


Example 6

M1- Multiplying matrices (ver. 6)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 3 & -5 & 2 \\ 0 & 1 & 0 & 2 \\ 1 & 1 & -4 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 2 & 1 \\ 2 & 5 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 1 & 0 & 3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & 2 & 1 \\ 2 & 5 & 2 \end{array}\right] \left[\begin{array}{rrrr} 1 & 3 & -5 & 2 \\ 0 & 1 & 0 & 2 \\ 1 & 1 & -4 & -1 \end{array}\right] = \left[\begin{array}{rrrr} 2 & 6 & -9 & 5 \\ 4 & 13 & -18 & 12 \end{array}\right]\]


Example 7

M1- Multiplying matrices (ver. 7)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 0 & -2 & 2 \\ -2 & 1 & 6 & -2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -3 & -1 & -6 \\ -2 & -1 & -5 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 2 & 3 & -3 \\ -2 & -3 & -2 & 5 \\ 0 & 1 & 5 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} -3 & -1 & -6 \\ -2 & -1 & -5 \end{array}\right] \left[\begin{array}{rrrr} 1 & 2 & 3 & -3 \\ -2 & -3 & -2 & 5 \\ 0 & 1 & 5 & -1 \end{array}\right] = \left[\begin{array}{rrrr} -1 & -9 & -37 & 10 \\ 0 & -6 & -29 & 6 \end{array}\right]\]


Example 8

M1- Multiplying matrices (ver. 8)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 5 & 6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 2 & -4 & -6 \\ -1 & -1 & 1 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & -1 & -3 & -1 \\ 1 & -1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 5 & 6 \end{array}\right] \left[\begin{array}{rrrr} 0 & -1 & -3 & -1 \\ 1 & -1 & -1 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] = \left[\begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 5 & -6 & -2 & 4 \end{array}\right]\]


Example 9

M1- Multiplying matrices (ver. 9)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -2 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 0 & -1 & 0 \\ 1 & 6 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 1 \\ 0 & 1 \\ 0 & -3 \\ 0 & 0 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 1 & 1 \\ 0 & 1 \\ 0 & -3 \\ 0 & 0 \end{array}\right] \left[\begin{array}{rrr} 0 & -1 & 0 \\ 1 & 6 & 2 \end{array}\right] = \left[\begin{array}{rrr} 1 & 5 & 2 \\ 1 & 6 & 2 \\ -3 & -18 & -6 \\ 0 & 0 & 0 \end{array}\right]\]


Example 10

M1- Multiplying matrices (ver. 10)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ -1 & -1 & -4 & -4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 2 & -1 & 3 & 3 \\ -1 & 1 & 1 & 0 \\ 1 & -1 & 0 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 0 \\ 0 & 1 \\ 2 & -5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -1 & 0 \\ 0 & 1 \\ 2 & -5 \end{array}\right] \left[\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ -1 & -1 & -4 & -4 \end{array}\right] = \left[\begin{array}{rrrr} 0 & -1 & -2 & -1 \\ -1 & -1 & -4 & -4 \\ 5 & 7 & 24 & 22 \end{array}\right]\]


Example 11

M1- Multiplying matrices (ver. 11)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -1 & -2 & 1 \\ -1 & -3 & -1 \\ -1 & -2 & 2 \\ 1 & 4 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 1 & 3 \\ 0 & 1 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 6 \\ -1 & 5 \\ -2 & 6 \\ 0 & -4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -1 & 6 \\ -1 & 5 \\ -2 & 6 \\ 0 & -4 \end{array}\right] \left[\begin{array}{rrr} 1 & 1 & 3 \\ 0 & 1 & 3 \end{array}\right] = \left[\begin{array}{rrr} -1 & 5 & 15 \\ -1 & 4 & 12 \\ -2 & 4 & 12 \\ 0 & -4 & -12 \end{array}\right]\]


Example 12

M1- Multiplying matrices (ver. 12)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -1 & -2 & -4 \\ 1 & 1 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & -2 & -2 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \\ -1 & -2 & -4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 2 & 3 & 1 \\ -1 & -1 & -1 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & 2 & 3 & 1 \\ -1 & -1 & -1 & -1 \end{array}\right] \left[\begin{array}{rrr} -1 & -2 & -2 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \\ -1 & -2 & -4 \end{array}\right] = \left[\begin{array}{rrr} -2 & -2 & 7 \\ 2 & 3 & 0 \end{array}\right]\]


Example 13

M1- Multiplying matrices (ver. 13)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 3 & 1 & 2 \\ 2 & 1 & 0 \\ 3 & 1 & 3 \\ 3 & 0 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 0 \\ -4 & 1 \\ 5 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -3 \\ 2 & -5 \\ 3 & -6 \\ 0 & 4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 3 & 1 & 2 \\ 2 & 1 & 0 \\ 3 & 1 & 3 \\ 3 & 0 & 4 \end{array}\right] \left[\begin{array}{rr} 1 & 0 \\ -4 & 1 \\ 5 & 3 \end{array}\right] = \left[\begin{array}{rr} 9 & 7 \\ -2 & 1 \\ 14 & 10 \\ 23 & 12 \end{array}\right]\]


Example 14

M1- Multiplying matrices (ver. 14)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & -3 & -3 & -3 \\ -1 & 4 & 5 & 2 \\ -1 & 4 & 6 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 6 & 0 \\ -5 & 1 \\ -5 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 0 & 2 \\ -4 & 5 \\ 0 & 4 \\ -1 & 3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & -3 & -3 & -3 \\ -1 & 4 & 5 & 2 \\ -1 & 4 & 6 & 1 \end{array}\right] \left[\begin{array}{rr} 0 & 2 \\ -4 & 5 \\ 0 & 4 \\ -1 & 3 \end{array}\right] = \left[\begin{array}{rr} 15 & -34 \\ -18 & 44 \\ -17 & 45 \end{array}\right]\]


Example 15

M1- Multiplying matrices (ver. 15)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 5 & 1 & 4 & -1 \\ 4 & 1 & 3 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 0 & -1 & 4 \\ 1 & 0 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} -2 & -4 & 1 & 0 \\ -1 & -4 & 4 & 5 \\ 1 & 3 & -2 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 0 & -1 & 4 \\ 1 & 0 & 1 \end{array}\right] \left[\begin{array}{rrrr} -2 & -4 & 1 & 0 \\ -1 & -4 & 4 & 5 \\ 1 & 3 & -2 & -2 \end{array}\right] = \left[\begin{array}{rrrr} 5 & 16 & -12 & -13 \\ -1 & -1 & -1 & -2 \end{array}\right]\]


Example 16

M1- Multiplying matrices (ver. 16)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -2 & -1 & 3 & 3 \\ 3 & 1 & -5 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & -1 & -2 \\ -1 & -2 & -3 \\ -1 & 1 & 1 \\ 0 & -2 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & -1 & 1 \\ -1 & -2 & 3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} -2 & -1 & 3 & 3 \\ 3 & 1 & -5 & -6 \end{array}\right] \left[\begin{array}{rrr} -1 & -1 & -2 \\ -1 & -2 & -3 \\ -1 & 1 & 1 \\ 0 & -2 & -6 \end{array}\right] = \left[\begin{array}{rrr} 0 & 1 & -8 \\ 1 & 2 & 22 \end{array}\right]\]


Example 17

M1- Multiplying matrices (ver. 17)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & 2 & 1 \\ -2 & -3 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 4 \\ 1 & 5 \\ 1 & 3 \\ 0 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 3 & -1 & 3 \\ 1 & -1 & 2 \\ -2 & -1 & 0 \\ -2 & 0 & -4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 1 & 4 \\ 1 & 5 \\ 1 & 3 \\ 0 & 2 \end{array}\right] \left[\begin{array}{rrr} 1 & 2 & 1 \\ -2 & -3 & -3 \end{array}\right] = \left[\begin{array}{rrr} -7 & -10 & -11 \\ -9 & -13 & -14 \\ -5 & -7 & -8 \\ -4 & -6 & -6 \end{array}\right]\]


Example 18

M1- Multiplying matrices (ver. 18)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 1 & -5 \\ 0 & 1 \\ 2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} -3 & 2 & -1 & -2 \\ -1 & 5 & -1 & -2 \\ -2 & 3 & -1 & -2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} -1 & 1 & -3 & 3 \\ 1 & -2 & 4 & -4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 1 & -5 \\ 0 & 1 \\ 2 & 1 \end{array}\right] \left[\begin{array}{rrrr} -1 & 1 & -3 & 3 \\ 1 & -2 & 4 & -4 \end{array}\right] = \left[\begin{array}{rrrr} -6 & 11 & -23 & 23 \\ 1 & -2 & 4 & -4 \\ -1 & 0 & -2 & 2 \end{array}\right]\]


Example 19

M1- Multiplying matrices (ver. 19)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -2 & 3 & -4 \\ 1 & -2 & 1 \\ 0 & -3 & -5 \\ -2 & 6 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & -1 & -1 \\ 3 & 1 & -4 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & 4 & 1 \\ -1 & 3 & 0 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & -1 & -1 \\ 3 & 1 & -4 & -6 \end{array}\right] \left[\begin{array}{rrr} -2 & 3 & -4 \\ 1 & -2 & 1 \\ 0 & -3 & -5 \\ -2 & 6 & 3 \end{array}\right] = \left[\begin{array}{rrr} 0 & 0 & -2 \\ 7 & -17 & -9 \end{array}\right]\]


Example 20

M1- Multiplying matrices (ver. 20)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 0 & 0 & -1 & -3 \\ 1 & 4 & 2 & 5 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 2 & 2 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & 3 & -1 \\ -2 & 5 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 0 & 0 & -1 & -3 \\ 1 & 4 & 2 & 5 \end{array}\right] \left[\begin{array}{rrr} 1 & 2 & 2 \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & -3 \end{array}\right] = \left[\begin{array}{rrr} 0 & 2 & 8 \\ 1 & 3 & -11 \end{array}\right]\]


Example 21

M1- Multiplying matrices (ver. 21)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 0 & 0 & 1 & 1 \\ 0 & 1 & -3 & -4 \\ -1 & 1 & -2 & -5 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 0 \\ -1 & -1 \\ 2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -3 & -5 \\ -4 & -3 \\ 1 & 3 \\ -4 & -6 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 0 & 0 & 1 & 1 \\ 0 & 1 & -3 & -4 \\ -1 & 1 & -2 & -5 \end{array}\right] \left[\begin{array}{rr} -3 & -5 \\ -4 & -3 \\ 1 & 3 \\ -4 & -6 \end{array}\right] = \left[\begin{array}{rr} -3 & -3 \\ 9 & 12 \\ 17 & 26 \end{array}\right]\]


Example 22

M1- Multiplying matrices (ver. 22)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} -4 & 4 \\ -2 & 5 \\ 5 & -6 \\ 4 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 2 & -4 & -1 & -4 \\ 2 & -5 & -1 & -5 \\ -1 & 1 & 0 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -5 \\ -1 & 6 \\ 0 & -3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 2 & -4 & -1 & -4 \\ 2 & -5 & -1 & -5 \\ -1 & 1 & 0 & 0 \end{array}\right] \left[\begin{array}{rr} -4 & 4 \\ -2 & 5 \\ 5 & -6 \\ 4 & -6 \end{array}\right] = \left[\begin{array}{rr} -21 & 18 \\ -23 & 19 \\ 2 & 1 \end{array}\right]\]


Example 23

M1- Multiplying matrices (ver. 23)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 4 & -1 & -3 & -4 \\ 0 & 1 & 4 & 6 \\ -1 & 0 & 0 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -3 & -2 \\ -1 & -3 \\ 2 & 3 \\ 0 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 0 \\ 1 & -1 \\ 5 & -5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 4 & -1 & -3 & -4 \\ 0 & 1 & 4 & 6 \\ -1 & 0 & 0 & 0 \end{array}\right] \left[\begin{array}{rr} -3 & -2 \\ -1 & -3 \\ 2 & 3 \\ 0 & -3 \end{array}\right] = \left[\begin{array}{rr} -17 & -2 \\ 7 & -9 \\ 3 & 2 \end{array}\right]\]


Example 24

M1- Multiplying matrices (ver. 24)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -2 & -1 & 0 & 2 \\ 3 & 1 & 1 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -1 & 2 \\ -2 & -1 & 4 \\ 0 & 1 & -3 \\ -3 & -2 & 6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -1 & -3 \\ -1 & 2 & 5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} -2 & -1 & 0 & 2 \\ 3 & 1 & 1 & -3 \end{array}\right] \left[\begin{array}{rrr} 1 & -1 & 2 \\ -2 & -1 & 4 \\ 0 & 1 & -3 \\ -3 & -2 & 6 \end{array}\right] = \left[\begin{array}{rrr} -6 & -1 & 4 \\ 10 & 3 & -11 \end{array}\right]\]


Example 25

M1- Multiplying matrices (ver. 25)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -1 & 2 & 5 & 1 \\ -1 & 1 & 4 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 3 & -5 & -4 \\ -1 & 2 & 2 \\ 1 & -2 & -1 \\ -4 & 6 & 3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} -1 & 2 & 5 & 1 \\ -1 & 1 & 4 & 2 \end{array}\right] \left[\begin{array}{rrr} 3 & -5 & -4 \\ -1 & 2 & 2 \\ 1 & -2 & -1 \\ -4 & 6 & 3 \end{array}\right] = \left[\begin{array}{rrr} -4 & 5 & 6 \\ -8 & 11 & 8 \end{array}\right]\]


Example 26

M1- Multiplying matrices (ver. 26)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -1 & -2 & -3 \\ -1 & -3 & -4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} -1 & 0 & 3 & 1 \\ 1 & 3 & 5 & 4 \\ -1 & -1 & 0 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & -1 & 1 & 0 \\ 0 & 1 & -2 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} -1 & -2 & -3 \\ -1 & -3 & -4 \end{array}\right] \left[\begin{array}{rrrr} -1 & 0 & 3 & 1 \\ 1 & 3 & 5 & 4 \\ -1 & -1 & 0 & -1 \end{array}\right] = \left[\begin{array}{rrrr} 2 & -3 & -13 & -6 \\ 2 & -5 & -18 & -9 \end{array}\right]\]


Example 27

M1- Multiplying matrices (ver. 27)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 1 & 0 \\ -1 & 4 \\ 0 & -1 \\ 2 & -5 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 4 \\ 1 & 5 \\ 1 & 5 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -1 & 3 \\ 0 & 1 & -5 \\ -3 & 2 & -3 \\ 3 & -2 & 5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -1 & 3 \\ 0 & 1 & -5 \\ -3 & 2 & -3 \\ 3 & -2 & 5 \end{array}\right] \left[\begin{array}{rr} 1 & 4 \\ 1 & 5 \\ 1 & 5 \end{array}\right] = \left[\begin{array}{rr} 3 & 14 \\ -4 & -20 \\ -4 & -17 \\ 6 & 27 \end{array}\right]\]


Example 28

M1- Multiplying matrices (ver. 28)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -2 & 6 & 5 & 3 \\ -1 & 3 & 2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -3 & 1 & 3 \\ 1 & 1 & -6 \\ 2 & -1 & -2 \\ 4 & -1 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & -3 & 6 \\ -1 & -3 & 5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} -2 & 6 & 5 & 3 \\ -1 & 3 & 2 & 1 \end{array}\right] \left[\begin{array}{rrr} -3 & 1 & 3 \\ 1 & 1 & -6 \\ 2 & -1 & -2 \\ 4 & -1 & -6 \end{array}\right] = \left[\begin{array}{rrr} 34 & -4 & -70 \\ 14 & -1 & -31 \end{array}\right]\]


Example 29

M1- Multiplying matrices (ver. 29)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 0 & 2 & -1 \\ -1 & 1 & -1 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & 1 & -1 \\ 1 & -2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & -2 & -5 \\ -1 & 1 & 1 & 2 \\ 0 & 3 & -2 & -6 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} -1 & 1 & -1 \\ 1 & -2 & 1 \end{array}\right] \left[\begin{array}{rrrr} 1 & 0 & -2 & -5 \\ -1 & 1 & 1 & 2 \\ 0 & 3 & -2 & -6 \end{array}\right] = \left[\begin{array}{rrrr} -2 & -2 & 5 & 13 \\ 3 & 1 & -6 & -15 \end{array}\right]\]


Example 30

M1- Multiplying matrices (ver. 30)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 0 & 1 & -1 \\ -1 & 4 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 4 & -2 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -1 & -2 \\ 0 & 1 & 5 \\ 0 & -1 & -4 \\ -1 & 2 & 4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & 4 & -2 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \left[\begin{array}{rrr} 1 & -1 & -2 \\ 0 & 1 & 5 \\ 0 & -1 & -4 \\ -1 & 2 & 4 \end{array}\right] = \left[\begin{array}{rrr} 0 & 7 & 30 \\ 0 & -1 & -4 \end{array}\right]\]


Example 31

M1- Multiplying matrices (ver. 31)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 0 & -1 & -4 \\ 1 & 1 & -1 \\ -1 & -1 & 2 \\ 1 & 0 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 0 & 1 \\ -1 & 1 \\ 2 & -2 \\ -4 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -3 & -4 \\ 4 & 5 \\ 2 & -3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 0 & -1 & -4 \\ 1 & 1 & -1 \\ -1 & -1 & 2 \\ 1 & 0 & -6 \end{array}\right] \left[\begin{array}{rr} -3 & -4 \\ 4 & 5 \\ 2 & -3 \end{array}\right] = \left[\begin{array}{rr} -12 & 7 \\ -1 & 4 \\ 3 & -7 \\ -15 & 14 \end{array}\right]\]


Example 32

M1- Multiplying matrices (ver. 32)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & -2 & -6 \\ 0 & 1 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -1 & -5 \\ 1 & 0 & -2 \\ 0 & -2 & -5 \\ 1 & 0 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & -1 & 0 & 2 \\ 1 & 1 & -3 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 0 & -1 & 0 & 2 \\ 1 & 1 & -3 & -2 \end{array}\right] \left[\begin{array}{rrr} 1 & -1 & -5 \\ 1 & 0 & -2 \\ 0 & -2 & -5 \\ 1 & 0 & -3 \end{array}\right] = \left[\begin{array}{rrr} 1 & 0 & -4 \\ 0 & 5 & 14 \end{array}\right]\]


Example 33

M1- Multiplying matrices (ver. 33)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 3 & 0 \\ 1 & 3 \\ 5 & 5 \\ -1 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & 3 & 5 \\ 0 & 1 & -2 & -2 \\ -1 & 1 & -4 & -5 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 2 \\ -1 & -1 \\ 5 & 5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 3 & 5 \\ 0 & 1 & -2 & -2 \\ -1 & 1 & -4 & -5 \end{array}\right] \left[\begin{array}{rr} 3 & 0 \\ 1 & 3 \\ 5 & 5 \\ -1 & 1 \end{array}\right] = \left[\begin{array}{rr} 13 & 20 \\ -7 & -9 \\ -17 & -22 \end{array}\right]\]


Example 34

M1- Multiplying matrices (ver. 34)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & -2 & -5 & 1 \\ 0 & 1 & 3 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 2 & -2 \\ 0 & 0 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -2 & -1 \\ 0 & 1 & 0 \\ -3 & 4 & 4 \\ 2 & -5 & -4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & -2 & -5 & 1 \\ 0 & 1 & 3 & 0 \end{array}\right] \left[\begin{array}{rrr} 1 & -2 & -1 \\ 0 & 1 & 0 \\ -3 & 4 & 4 \\ 2 & -5 & -4 \end{array}\right] = \left[\begin{array}{rrr} 18 & -29 & -25 \\ -9 & 13 & 12 \end{array}\right]\]


Example 35

M1- Multiplying matrices (ver. 35)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 1 & 2 \\ 2 & 5 \\ -2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 5 & -1 \\ -4 & 1 \\ 2 & 0 \\ -5 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -5 & -5 \\ 0 & 1 & 2 \\ -3 & 4 & -6 \\ -3 & 4 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -5 & -5 \\ 0 & 1 & 2 \\ -3 & 4 & -6 \\ -3 & 4 & -2 \end{array}\right] \left[\begin{array}{rr} 1 & 2 \\ 2 & 5 \\ -2 & 1 \end{array}\right] = \left[\begin{array}{rr} 1 & -28 \\ -2 & 7 \\ 17 & 8 \\ 9 & 12 \end{array}\right]\]


Example 36

M1- Multiplying matrices (ver. 36)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 3 \\ -2 & -1 & 2 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -3 \\ -1 & 4 \\ 1 & -5 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} -2 & 1 & -1 & 2 \\ 3 & -2 & 0 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 1 & -3 \\ -1 & 4 \\ 1 & -5 \end{array}\right] \left[\begin{array}{rrrr} -2 & 1 & -1 & 2 \\ 3 & -2 & 0 & -2 \end{array}\right] = \left[\begin{array}{rrrr} -11 & 7 & -1 & 8 \\ 14 & -9 & 1 & -10 \\ -17 & 11 & -1 & 12 \end{array}\right]\]


Example 37

M1- Multiplying matrices (ver. 37)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -2 & -1 & 3 \\ -3 & -2 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 3 & -2 & 0 & 0 \\ 2 & -1 & 0 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & 0 & -1 & -2 \\ 1 & -1 & -1 & 1 \\ 1 & -2 & -3 & -2 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} -2 & -1 & 3 \\ -3 & -2 & 4 \end{array}\right] \left[\begin{array}{rrrr} 0 & 0 & -1 & -2 \\ 1 & -1 & -1 & 1 \\ 1 & -2 & -3 & -2 \end{array}\right] = \left[\begin{array}{rrrr} 2 & -5 & -6 & -3 \\ 2 & -6 & -7 & -4 \end{array}\right]\]


Example 38

M1- Multiplying matrices (ver. 38)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} -4 & 0 \\ 3 & 4 \\ -5 & -1 \\ 3 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -3 \\ 0 & 1 \\ -2 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & -1 & -1 & 0 \\ 0 & 1 & 0 & 2 \\ 1 & -2 & -2 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 0 & -1 & -1 & 0 \\ 0 & 1 & 0 & 2 \\ 1 & -2 & -2 & -1 \end{array}\right] \left[\begin{array}{rr} -4 & 0 \\ 3 & 4 \\ -5 & -1 \\ 3 & 0 \end{array}\right] = \left[\begin{array}{rr} 2 & -3 \\ 9 & 4 \\ -3 & -6 \end{array}\right]\]


Example 39

M1- Multiplying matrices (ver. 39)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & -2 & -3 & 1 \\ 1 & -1 & 1 & -1 \\ -1 & 2 & 4 & -2 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -1 \\ -3 & 4 \\ -2 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 5 & -4 & 3 & -1 \\ 4 & -3 & 2 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 1 & -1 \\ -3 & 4 \\ -2 & -3 \end{array}\right] \left[\begin{array}{rrrr} 5 & -4 & 3 & -1 \\ 4 & -3 & 2 & -1 \end{array}\right] = \left[\begin{array}{rrrr} 1 & -1 & 1 & 0 \\ 1 & 0 & -1 & -1 \\ -22 & 17 & -12 & 5 \end{array}\right]\]


Example 40

M1- Multiplying matrices (ver. 40)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 2 & 2 & 5 \\ 0 & 1 & -4 \\ -1 & 0 & -4 \\ -1 & -2 & 0 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -5 & -1 \\ 0 & 0 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -2 & 5 \\ 1 & -3 \\ 1 & -4 \\ 1 & -3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -2 & 5 \\ 1 & -3 \\ 1 & -4 \\ 1 & -3 \end{array}\right] \left[\begin{array}{rrr} 1 & -5 & -1 \\ 0 & 0 & 1 \end{array}\right] = \left[\begin{array}{rrr} -2 & 10 & 7 \\ 1 & -5 & -4 \\ 1 & -5 & -5 \\ 1 & -5 & -4 \end{array}\right]\]


Example 41

M1- Multiplying matrices (ver. 41)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & 2 & -3 \\ 0 & 1 & 0 \\ 0 & 5 & 1 \\ 1 & 2 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -3 & -1 \\ 2 & -5 \\ -2 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & 2 \\ -2 & 3 \\ 2 & 3 \\ -5 & 1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & 2 & -3 \\ 0 & 1 & 0 \\ 0 & 5 & 1 \\ 1 & 2 & 2 \end{array}\right] \left[\begin{array}{rr} -3 & -1 \\ 2 & -5 \\ -2 & 1 \end{array}\right] = \left[\begin{array}{rr} 7 & -14 \\ 2 & -5 \\ 8 & -24 \\ -3 & -9 \end{array}\right]\]


Example 42

M1- Multiplying matrices (ver. 42)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & 2 & 0 \\ -3 & -5 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -1 & 0 & 6 \\ -1 & 1 & 1 \\ -1 & 0 & 5 \\ 3 & -3 & -6 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & -1 & 0 \\ -1 & 1 & 0 & -4 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & -1 & 0 \\ -1 & 1 & 0 & -4 \end{array}\right] \left[\begin{array}{rrr} -1 & 0 & 6 \\ -1 & 1 & 1 \\ -1 & 0 & 5 \\ 3 & -3 & -6 \end{array}\right] = \left[\begin{array}{rrr} 0 & 0 & 1 \\ -12 & 13 & 19 \end{array}\right]\]


Example 43

M1- Multiplying matrices (ver. 43)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 2 & -2 \\ 3 & -5 \\ 1 & 0 \\ -3 & 6 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 0 & 1 \\ -1 & 6 \\ -1 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -4 & 6 \\ -1 & -3 & -5 \\ 0 & -4 & -3 \\ 0 & -4 & 0 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -4 & 6 \\ -1 & -3 & -5 \\ 0 & -4 & -3 \\ 0 & -4 & 0 \end{array}\right] \left[\begin{array}{rr} 0 & 1 \\ -1 & 6 \\ -1 & 2 \end{array}\right] = \left[\begin{array}{rr} -2 & -11 \\ 8 & -29 \\ 7 & -30 \\ 4 & -24 \end{array}\right]\]


Example 44

M1- Multiplying matrices (ver. 44)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & -2 & 1 & 5 \\ 0 & 1 & -3 & -4 \\ -1 & 0 & 6 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & 5 \\ 0 & 1 \\ -1 & 0 \\ -1 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 5 & 0 \\ -5 & 6 \\ -4 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 1 & -2 & 1 & 5 \\ 0 & 1 & -3 & -4 \\ -1 & 0 & 6 & 4 \end{array}\right] \left[\begin{array}{rr} 1 & 5 \\ 0 & 1 \\ -1 & 0 \\ -1 & -3 \end{array}\right] = \left[\begin{array}{rr} -5 & -12 \\ 7 & 13 \\ -11 & -17 \end{array}\right]\]


Example 45

M1- Multiplying matrices (ver. 45)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} 1 & 0 & -2 & -5 \\ 2 & 1 & -2 & -5 \\ 0 & -1 & -1 & -2 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -6 & 4 \\ 0 & 1 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 1 & 0 & 2 & 2 \\ -1 & 1 & 0 & -5 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -6 & 4 \\ 0 & 1 & -1 \end{array}\right] \left[\begin{array}{rrrr} 1 & 0 & -2 & -5 \\ 2 & 1 & -2 & -5 \\ 0 & -1 & -1 & -2 \end{array}\right] = \left[\begin{array}{rrrr} -11 & -10 & 6 & 17 \\ 2 & 2 & -1 & -3 \end{array}\right]\]


Example 46

M1- Multiplying matrices (ver. 46)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} 1 & 5 & 3 \\ 1 & 6 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 2 & -1 & -6 \\ 5 & 1 & -5 \\ 1 & -1 & -6 \\ -1 & 1 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 3 & 4 & -5 & 2 \\ 2 & 3 & -4 & 1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrrr} 3 & 4 & -5 & 2 \\ 2 & 3 & -4 & 1 \end{array}\right] \left[\begin{array}{rrr} 2 & -1 & -6 \\ 5 & 1 & -5 \\ 1 & -1 & -6 \\ -1 & 1 & 4 \end{array}\right] = \left[\begin{array}{rrr} 19 & 8 & 0 \\ 14 & 6 & 1 \end{array}\right]\]


Example 47

M1- Multiplying matrices (ver. 47)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrrr} -2 & 1 & -1 & 2 \\ 3 & -2 & 4 & -5 \\ -3 & 2 & -3 & 4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrrr} 0 & 1 & 1 & 3 \\ -1 & 0 & 0 & -1 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -1 & -3 \\ -1 & 3 \\ -2 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -1 & -3 \\ -1 & 3 \\ -2 & -1 \end{array}\right] \left[\begin{array}{rrrr} 0 & 1 & 1 & 3 \\ -1 & 0 & 0 & -1 \end{array}\right] = \left[\begin{array}{rrrr} 3 & -1 & -1 & 0 \\ -3 & -1 & -1 & -6 \\ 1 & -2 & -2 & -5 \end{array}\right]\]


Example 48

M1- Multiplying matrices (ver. 48)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rr} 1 & 1 \\ 2 & 3 \\ 3 & 4 \\ -4 & -3 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & 3 \\ 0 & -1 & -2 \\ -1 & 2 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 1 & -3 \\ 0 & 1 \\ 0 & -1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & 3 \\ 0 & -1 & -2 \\ -1 & 2 & 3 \end{array}\right] \left[\begin{array}{rr} 1 & -3 \\ 0 & 1 \\ 0 & -1 \end{array}\right] = \left[\begin{array}{rr} 1 & -3 \\ 0 & -2 \\ 0 & 1 \\ -1 & 2 \end{array}\right]\]


Example 49

M1- Multiplying matrices (ver. 49)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -1 & 0 & 1 \\ 2 & -1 & -4 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} 1 & 5 & 0 \\ 1 & 6 & -1 \\ -2 & -5 & -4 \\ -1 & -5 & 3 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} 5 & 3 \\ 3 & -2 \\ -2 & -2 \\ 4 & 1 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} 5 & 3 \\ 3 & -2 \\ -2 & -2 \\ 4 & 1 \end{array}\right] \left[\begin{array}{rrr} -1 & 0 & 1 \\ 2 & -1 & -4 \end{array}\right] = \left[\begin{array}{rrr} 1 & -3 & -7 \\ -7 & 2 & 11 \\ -2 & 2 & 6 \\ -2 & -1 & 0 \end{array}\right]\]


Example 50

M1- Multiplying matrices (ver. 50)

Of the following three matrices, only two may be multiplied.

\[\left[\begin{array}{rrr} -1 & 3 & 3 \\ -1 & 3 & 2 \end{array}\right] \hspace{2em} \left[\begin{array}{rr} -5 & -2 \\ -2 & -1 \\ 1 & 0 \\ 3 & 1 \end{array}\right] \hspace{2em} \left[\begin{array}{rrr} -2 & -1 & -4 \\ -5 & 1 & 5 \\ -3 & -1 & -4 \\ -4 & -1 & -3 \end{array}\right]\]

Explain which two may be multiplied and why. Then show how to find their product.

Answer.

\[\left[\begin{array}{rr} -5 & -2 \\ -2 & -1 \\ 1 & 0 \\ 3 & 1 \end{array}\right] \left[\begin{array}{rrr} -1 & 3 & 3 \\ -1 & 3 & 2 \end{array}\right] = \left[\begin{array}{rrr} 7 & -21 & -19 \\ 3 & -9 & -8 \\ -1 & 3 & 3 \\ -4 & 12 & 11 \end{array}\right]\]