M3- Computing matrix inverses


Example 1

M3- Computing matrix inverses (ver. 1)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 2 & 8 & -1 \\ -1 & -3 & 1 \\ -1 & -4 & 1 \end{array}\right]\]


Example 2

M3- Computing matrix inverses (ver. 2)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 3

M3- Computing matrix inverses (ver. 3)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & 4 & -4 \\ 2 & 4 & -3 \\ 0 & -1 & 1 \end{array}\right]\]


Example 4

M3- Computing matrix inverses (ver. 4)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 5

M3- Computing matrix inverses (ver. 5)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 6

M3- Computing matrix inverses (ver. 6)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 21 & -6 & -4 \\ -10 & 3 & 2 \\ -4 & 1 & 1 \end{array}\right]\]


Example 7

M3- Computing matrix inverses (ver. 7)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -6 & 4 & 3 \\ -1 & 1 & 1 \\ 4 & -3 & -3 \end{array}\right]\]


Example 8

M3- Computing matrix inverses (ver. 8)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -28 & 1 & -5 \\ -6 & 0 & -1 \\ 5 & 0 & 1 \end{array}\right]\]


Example 9

M3- Computing matrix inverses (ver. 9)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & 8 & 2 \\ 0 & -2 & -1 \\ 0 & 3 & 1 \end{array}\right]\]


Example 10

M3- Computing matrix inverses (ver. 10)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -1 & 0 & 0 \\ -2 & -4 & -1 \\ -2 & -5 & -1 \end{array}\right]\]


Example 11

M3- Computing matrix inverses (ver. 11)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -1 & 0 & 0 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{array}\right]\]


Example 12

M3- Computing matrix inverses (ver. 12)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -4 & -5 & 3 \\ 1 & 1 & -1 \\ 1 & 1 & 0 \end{array}\right]\]


Example 13

M3- Computing matrix inverses (ver. 13)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array}\right]\]


Example 14

M3- Computing matrix inverses (ver. 14)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 5 & -4 & 9 \\ -6 & 5 & -10 \\ 1 & -1 & 2 \end{array}\right]\]


Example 15

M3- Computing matrix inverses (ver. 15)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & 2 & -2 \\ -4 & -5 & 3 \\ -2 & -2 & 1 \end{array}\right]\]


Example 16

M3- Computing matrix inverses (ver. 16)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} -1 & 1 & -6 \\ -1 & 0 & -1 \\ 1 & 1 & -3 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & -3 & -1 \\ -4 & 9 & 5 \\ -1 & 2 & 1 \end{array}\right]\]


Example 17

M3- Computing matrix inverses (ver. 17)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 18

M3- Computing matrix inverses (ver. 18)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -1 & -6 & 1 \\ 2 & 12 & -1 \\ -2 & -11 & 1 \end{array}\right]\]


Example 19

M3- Computing matrix inverses (ver. 19)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 6 & -13 & -5 \\ 6 & -12 & -5 \\ -1 & 2 & 1 \end{array}\right]\]


Example 20

M3- Computing matrix inverses (ver. 20)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 21

M3- Computing matrix inverses (ver. 21)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 6 & -20 & 5 \\ 3 & -11 & 3 \\ 1 & -4 & 1 \end{array}\right]\]


Example 22

M3- Computing matrix inverses (ver. 22)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 4 & -1 & 3 \\ 4 & 1 & -4 \\ -3 & 1 & -3 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 24 & -3 & 28 \\ 7 & -1 & 8 \end{array}\right]\]


Example 23

M3- Computing matrix inverses (ver. 23)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -5 & 13 & 4 \\ -5 & 12 & 4 \\ -1 & 2 & 1 \end{array}\right]\]


Example 24

M3- Computing matrix inverses (ver. 24)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 25

M3- Computing matrix inverses (ver. 25)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -7 & 15 & -3 \\ -12 & 25 & -5 \\ 2 & -4 & 1 \end{array}\right]\]


Example 26

M3- Computing matrix inverses (ver. 26)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 13 & -4 & 10 \\ -20 & 6 & -15 \\ -4 & 1 & -3 \end{array}\right]\]


Example 27

M3- Computing matrix inverses (ver. 27)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -8 & -2 & 5 \\ -13 & -4 & 8 \\ 2 & 1 & -1 \end{array}\right]\]


Example 28

M3- Computing matrix inverses (ver. 28)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 29

M3- Computing matrix inverses (ver. 29)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} -5 & -3 & 1 \\ -3 & -2 & 1 \\ 1 & 2 & -2 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 2 & -4 & -1 \\ -5 & 9 & 2 \\ -4 & 7 & 1 \end{array}\right]\]


Example 30

M3- Computing matrix inverses (ver. 30)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 31

M3- Computing matrix inverses (ver. 31)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 14 & 13 & -2 \\ 5 & 5 & -1 \\ -4 & -4 & 1 \end{array}\right]\]


Example 32

M3- Computing matrix inverses (ver. 32)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 33

M3- Computing matrix inverses (ver. 33)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 3 & -1 & -4 \\ 0 & 1 & 3 \\ 1 & -1 & -3 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 0 & 1 & 1 \\ 3 & -5 & -9 \\ -1 & 2 & 3 \end{array}\right]\]


Example 34

M3- Computing matrix inverses (ver. 34)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 0 & -3 & -4 \\ -1 & 3 & 2 \\ 0 & 1 & 1 \end{array}\right]\]


Example 35

M3- Computing matrix inverses (ver. 35)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -2 & 3 & -6 \\ -1 & 1 & -1 \\ 2 & -2 & 3 \end{array}\right]\]


Example 36

M3- Computing matrix inverses (ver. 36)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 5 & 5 & 1 \\ 1 & 1 & 0 \\ 1 & 2 & 1 \end{array}\right]\]


Example 37

M3- Computing matrix inverses (ver. 37)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 1 & -1 & -4 \\ 1 & 0 & 2 \\ 0 & 0 & 1 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 0 & 1 & -2 \\ -1 & 1 & -6 \\ 0 & 0 & 1 \end{array}\right]\]


Example 38

M3- Computing matrix inverses (ver. 38)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & -3 & 3 \\ 0 & -4 & 5 \\ 0 & -1 & 1 \end{array}\right]\]


Example 39

M3- Computing matrix inverses (ver. 39)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 2 \\ -2 & -1 & 5 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 7 & -11 & 3 \\ -4 & 7 & -2 \\ 2 & -3 & 1 \end{array}\right]\]


Example 40

M3- Computing matrix inverses (ver. 40)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -5 & -2 & 12 \\ -7 & -3 & 18 \\ -3 & -1 & 7 \end{array}\right]\]


Example 41

M3- Computing matrix inverses (ver. 41)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 1 & 0 & -5 \\ 0 & 1 & -4 \\ 0 & 0 & 1 \end{array}\right]\]


Example 42

M3- Computing matrix inverses (ver. 42)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -30 & -3 & -4 \\ -22 & -2 & -3 \\ 9 & 1 & 1 \end{array}\right]\]


Example 43

M3- Computing matrix inverses (ver. 43)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} -3 & 0 & 4 \\ 2 & 0 & -3 \\ 1 & -1 & -1 \end{array}\right]\]


Example 44

M3- Computing matrix inverses (ver. 44)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 6 & 1 & -13 \\ 5 & 1 & -10 \\ 0 & 0 & 1 \end{array}\right]\]


Example 45

M3- Computing matrix inverses (ver. 45)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 4 & 1 & -1 \\ -3 & -1 & 3 \end{array}\right]\]

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 2 & -1 & -1 \\ -9 & 6 & 5 \\ -1 & 1 & 1 \end{array}\right]\]


Example 46

M3- Computing matrix inverses (ver. 46)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 4 & 17 & 1 \\ -1 & -4 & 0 \\ -1 & -5 & 0 \end{array}\right]\]


Example 47

M3- Computing matrix inverses (ver. 47)

Show how to compute the inverse of the following invertible matrix.

\[A=\left[\begin{array}{rrr} 2 & -1 & 2 \\ 0 & 1 & -5 \\ 1 & 0 & -1 \end{array}\right]\]

Answer.

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


Example 48

M3- Computing matrix inverses (ver. 48)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

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


Example 49

M3- Computing matrix inverses (ver. 49)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 19 & 1 & 3 \\ 18 & 1 & 3 \\ 5 & 0 & 1 \end{array}\right]\]


Example 50

M3- Computing matrix inverses (ver. 50)

Show how to compute the inverse of the following invertible matrix.

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

Answer.

\[A^{-1}=\left[\begin{array}{rrr} 19 & 4 & -33 \\ -14 & -3 & 24 \\ 4 & 1 & -7 \end{array}\right]\]