## M4 - Finding a matrix inverse

#### Example 1 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} -3 & 5 & -3 & 0 \\ 4 & -7 & 4 & 1 \\ 0 & 2 & 1 & -6 \\ -3 & 5 & 1 & 1 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} -180 & -131 & -21 & 5 \\ -103 & -75 & -12 & 3 \\ 8 & 6 & 1 & 0 \\ -33 & -24 & -4 & 1 \end{array}\right]$

#### Example 2 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 5 & 8 & 1 & -1 \\ 2 & 5 & 3 & -8 \\ -2 & -3 & 0 & -1 \\ 4 & 3 & 0 & -6 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} -63 & 21 & -129 & 4 \\ 48 & -16 & 98 & -3 \\ -86 & 29 & -176 & 5 \\ -18 & 6 & -37 & 1 \end{array}\right]$

#### Example 3 🔗

Show how to find the inverse of the matrix $$P= \left[\begin{array}{cccc} 1 & 5 & 7 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 2 & -2 \\ 0 & 0 & 1 & -2 \end{array}\right]$$.

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

#### Example 4 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 1 & 1 & 0 & -5 \\ 2 & -1 & 5 & 3 \\ 0 & 1 & -2 & -5 \\ -1 & 1 & -2 & 0 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} -15 & 10 & 21 & 4 \\ 1 & 0 & -1 & 1 \\ 8 & -5 & -11 & -2 \\ -3 & 2 & 4 & 1 \end{array}\right]$

#### Example 5 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} -2 & -3 & 2 & 5 \\ 4 & 1 & -1 & 0 \\ 3 & 3 & -2 & -4 \\ 0 & -3 & 2 & 8 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 4 & 0 & 3 & -1 \\ 8 & -2 & 8 & -1 \\ 24 & -3 & 20 & -5 \\ -3 & 0 & -2 & 1 \end{array}\right]$

#### Example 6 🔗

Show how to find the inverse of the matrix $$A= \left[\begin{array}{cccc} 1 & -2 & 0 & -6 \\ 1 & -1 & 1 & -6 \\ -1 & 2 & 1 & 5 \\ 0 & 3 & 0 & 4 \end{array}\right]$$.

$A^{-1}= \left[\begin{array}{cccc} 21 & -10 & 10 & 4 \\ -8 & 4 & -4 & -1 \\ 7 & -3 & 4 & 1 \\ 6 & -3 & 3 & 1 \end{array}\right]$

#### Example 7 🔗

Show how to find the inverse of the matrix $$P= \left[\begin{array}{cccc} -1 & -1 & 1 & -2 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & -3 \\ 1 & 1 & 0 & 0 \end{array}\right]$$.

$P^{-1}= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ 3 & 2 & -2 & 3 \\ 1 & 1 & -1 & 1 \end{array}\right]$

#### Example 8 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 0 & 2 & -4 & -5 \\ 0 & 1 & -4 & -2 \\ -2 & 0 & 7 & 8 \\ 1 & -2 & 1 & 1 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} 35 & 8 & 19 & 39 \\ 22 & 5 & 12 & 24 \\ 2 & 0 & 1 & 2 \\ 7 & 2 & 4 & 8 \end{array}\right]$

#### Example 9 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 1 & 2 & 3 & -8 \\ -1 & -1 & -1 & 0 \\ 0 & -1 & -1 & 5 \\ 0 & 0 & -2 & 7 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -10 & -11 & -9 & -5 \\ 3 & 3 & 2 & 2 \\ 7 & 7 & 7 & 3 \\ 2 & 2 & 2 & 1 \end{array}\right]$

#### Example 10 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 1 & -5 & -4 & -7 \\ 0 & 1 & 0 & 2 \\ -1 & 4 & 5 & 3 \\ 0 & -3 & 0 & -5 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 5 & 24 & 4 & 5 \\ 0 & -5 & 0 & -2 \\ 1 & 7 & 1 & 2 \\ 0 & 3 & 0 & 1 \end{array}\right]$

#### Example 11 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 1 & -3 & 8 & -2 \\ 0 & 1 & -3 & -1 \\ 0 & 2 & -5 & -1 \\ 2 & 5 & -8 & -5 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -7 & 29 & -35 & 4 \\ 4 & -19 & 21 & -2 \\ 2 & -9 & 10 & -1 \\ -2 & 7 & -9 & 1 \end{array}\right]$

#### Example 12 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 7 & -4 & 5 & 5 \\ -1 & 1 & 0 & 3 \\ 3 & -2 & 2 & 1 \\ -5 & 1 & -4 & -8 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 6 & -3 & -13 & 1 \\ -6 & 1 & 11 & -2 \\ -17 & 6 & 35 & -4 \\ 4 & -1 & -8 & 1 \end{array}\right]$

#### Example 13 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 1 & 3 & -1 & -7 \\ -1 & -2 & 0 & 2 \\ 1 & 4 & -1 & -7 \\ 0 & -2 & 1 & 6 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} 2 & 1 & 0 & 2 \\ -1 & 0 & 1 & 0 \\ -2 & -6 & -4 & -5 \\ 0 & 1 & 1 & 1 \end{array}\right]$

#### Example 14 🔗

Show how to find the inverse of the matrix $$P= \left[\begin{array}{cccc} -1 & 2 & 0 & 4 \\ 2 & -5 & -1 & -6 \\ -2 & 8 & 5 & -2 \\ 1 & -7 & -4 & 5 \end{array}\right]$$.

$P^{-1}= \left[\begin{array}{cccc} -77 & -46 & -6 & 4 \\ -8 & -5 & -1 & 0 \\ -24 & -14 & -1 & 2 \\ -15 & -9 & -1 & 1 \end{array}\right]$

#### Example 15 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} -1 & 2 & -3 & 8 \\ -1 & 1 & -3 & 3 \\ 1 & -1 & 4 & -2 \\ 0 & -1 & -4 & -8 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} 2 & -2 & 1 & 1 \\ -4 & -16 & -20 & -5 \\ -1 & -2 & -3 & -1 \\ 1 & 3 & 4 & 1 \end{array}\right]$

#### Example 16 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 1 & 1 & -1 & -5 \\ -3 & 4 & -7 & -6 \\ 0 & -2 & 3 & 6 \\ 0 & -2 & 5 & 7 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -26 & -9 & -33 & 2 \\ -27 & -9 & -35 & 3 \\ 6 & 2 & 7 & 0 \\ -12 & -4 & -15 & 1 \end{array}\right]$

#### Example 17 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 2 & 3 & -2 & 3 \\ -1 & -1 & 1 & -1 \\ 1 & 4 & 0 & 0 \\ 4 & 7 & -3 & 4 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} -4 & 4 & -3 & 4 \\ 1 & -1 & 1 & -1 \\ -3 & 7 & -3 & 4 \\ 0 & 3 & -1 & 1 \end{array}\right]$

#### Example 18 🔗

Show how to find the inverse of the matrix $$C= \left[\begin{array}{cccc} 0 & -1 & -1 & -7 \\ 0 & 1 & 0 & 2 \\ 1 & -4 & 2 & -1 \\ -1 & 4 & -2 & 2 \end{array}\right]$$.

$C^{-1}= \left[\begin{array}{cccc} 2 & 6 & 4 & 3 \\ 0 & 1 & -2 & -2 \\ -1 & -1 & -5 & -5 \\ 0 & 0 & 1 & 1 \end{array}\right]$

#### Example 19 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} -3 & 8 & -1 & -8 \\ -2 & 5 & -1 & -6 \\ -2 & 5 & 0 & -5 \\ 1 & -4 & 4 & 6 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -10 & -10 & 22 & -5 \\ -1 & -3 & 4 & -1 \\ -3 & -2 & 6 & -1 \\ 3 & 1 & -5 & 1 \end{array}\right]$

#### Example 20 🔗

Show how to find the inverse of the matrix $$A= \left[\begin{array}{cccc} 1 & 8 & 0 & 1 \\ -1 & -2 & 1 & -5 \\ 1 & 3 & 0 & 0 \\ 0 & 5 & 1 & -4 \end{array}\right]$$.

$A^{-1}= \left[\begin{array}{cccc} -3 & -3 & 1 & 3 \\ 1 & 1 & 0 & -1 \\ -21 & -25 & -4 & 26 \\ -4 & -5 & -1 & 5 \end{array}\right]$

#### Example 21 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 1 & 1 & 2 & -3 \\ 0 & 1 & 5 & -5 \\ 0 & 1 & 6 & -6 \\ 0 & 1 & 4 & -3 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 1 & -6 & 4 & 1 \\ 0 & 6 & -5 & 0 \\ 0 & -3 & 2 & 1 \\ 0 & -2 & 1 & 1 \end{array}\right]$

#### Example 22 🔗

Show how to find the inverse of the matrix $$P= \left[\begin{array}{cccc} -1 & -2 & -3 & 6 \\ 0 & 1 & 2 & -5 \\ 0 & 0 & 1 & -2 \\ 1 & 0 & 0 & 1 \end{array}\right]$$.

$P^{-1}= \left[\begin{array}{cccc} 1 & 2 & -1 & 2 \\ -1 & -1 & -1 & -1 \\ -2 & -4 & 3 & -2 \\ -1 & -2 & 1 & -1 \end{array}\right]$

#### Example 23 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} 1 & -3 & -7 & 5 \\ 1 & -2 & -4 & 6 \\ 0 & 2 & 7 & 4 \\ 1 & -1 & -2 & 6 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} -18 & 23 & -6 & -4 \\ 8 & -13 & 2 & 5 \\ -4 & 6 & -1 & -2 \\ 3 & -4 & 1 & 1 \end{array}\right]$

#### Example 24 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} -2 & -2 & 1 & -5 \\ -3 & -2 & 6 & 2 \\ 1 & 1 & -1 & 2 \\ -1 & 0 & 2 & 5 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -8 & -3 & -22 & 2 \\ 13 & 6 & 39 & -5 \\ 1 & 1 & 4 & -1 \\ -2 & -1 & -6 & 1 \end{array}\right]$

#### Example 25 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 1 & -8 \\ -4 & 0 & 1 & 8 \\ -5 & 0 & 2 & 8 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} -8 & 0 & -6 & 3 \\ -16 & 1 & -9 & 4 \\ -8 & 0 & -7 & 4 \\ -3 & 0 & -2 & 1 \end{array}\right]$

#### Example 26 🔗

Show how to find the inverse of the matrix $$A= \left[\begin{array}{cccc} -1 & 3 & -3 & 1 \\ -2 & 5 & -6 & 4 \\ -1 & 2 & -2 & -1 \\ 0 & 2 & -1 & 1 \end{array}\right]$$.

$A^{-1}= \left[\begin{array}{cccc} 17 & -5 & -8 & -5 \\ -4 & 1 & 2 & 2 \\ -11 & 3 & 5 & 4 \\ -3 & 1 & 1 & 1 \end{array}\right]$

#### Example 27 🔗

Show how to find the inverse of the matrix $$C= \left[\begin{array}{cccc} 1 & -7 & 7 & -4 \\ -1 & 5 & -5 & 3 \\ 2 & -3 & 4 & 2 \\ 0 & -3 & 3 & -2 \end{array}\right]$$.

$C^{-1}= \left[\begin{array}{cccc} -1 & -2 & 0 & -1 \\ -12 & -10 & 1 & 10 \\ -10 & -8 & 1 & 9 \\ 3 & 3 & 0 & -2 \end{array}\right]$

#### Example 28 🔗

Show how to find the inverse of the matrix $$P= \left[\begin{array}{cccc} -1 & 2 & -1 & 6 \\ -1 & 1 & -1 & 6 \\ -1 & -2 & 0 & 3 \\ 0 & -3 & 0 & 1 \end{array}\right]$$.

$P^{-1}= \left[\begin{array}{cccc} 7 & -7 & -1 & 3 \\ 1 & -1 & 0 & 0 \\ 12 & -13 & 1 & 3 \\ 3 & -3 & 0 & 1 \end{array}\right]$

#### Example 29 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 1 & 3 & 4 & -5 \\ 1 & 4 & 5 & -4 \\ -1 & -1 & -1 & 4 \\ 1 & 0 & 0 & -4 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} -4 & 4 & 4 & 5 \\ 0 & -1 & -5 & -4 \\ 0 & 1 & 4 & 3 \\ -1 & 1 & 1 & 1 \end{array}\right]$

#### Example 30 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 5 \\ 1 & 1 & -3 & -2 \\ 0 & 1 & -2 & -4 \\ -2 & -1 & 1 & -8 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} -14 & 5 & -10 & -5 \\ -4 & 0 & -1 & -2 \\ -8 & 2 & -5 & -3 \\ 3 & -1 & 2 & 1 \end{array}\right]$

#### Example 31 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 2 & 7 & 6 & 6 \\ 1 & 6 & 5 & 7 \\ 1 & 4 & 4 & 3 \\ -1 & -2 & -2 & 0 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} 6 & -6 & 2 & 7 \\ -5 & 6 & -4 & -8 \\ 2 & -3 & 3 & 4 \\ 2 & -2 & 1 & 3 \end{array}\right]$

#### Example 32 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} -2 & 1 & 4 & -5 \\ 1 & 2 & -6 & 8 \\ 1 & 0 & -3 & 4 \\ 1 & 5 & -7 & 8 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} 8 & -9 & 24 & 2 \\ 4 & -4 & 11 & 1 \\ 12 & -11 & 33 & 2 \\ 7 & -6 & 19 & 1 \end{array}\right]$

#### Example 33 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} -1 & -2 & 7 & 2 \\ 1 & 1 & 0 & 2 \\ -1 & -1 & 1 & -1 \\ 1 & -1 & 6 & 3 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} -1 & 0 & 1 & 1 \\ 5 & -9 & -17 & -3 \\ 2 & -4 & -7 & -1 \\ -2 & 5 & 8 & 1 \end{array}\right]$

#### Example 34 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} -1 & 1 & -1 & -4 \\ -1 & 0 & 1 & 2 \\ 0 & 1 & -1 & -1 \\ 0 & 1 & -1 & 0 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} -1 & 0 & 4 & -3 \\ -1 & 1 & 6 & -4 \\ -1 & 1 & 6 & -5 \\ 0 & 0 & -1 & 1 \end{array}\right]$

#### Example 35 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 0 & 0 & -1 & 5 \\ -1 & 0 & 0 & 2 \\ 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 4 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} 0 & -3 & -2 & 2 \\ 0 & 4 & 4 & -3 \\ -1 & -5 & -5 & 5 \\ 0 & -1 & -1 & 1 \end{array}\right]$

#### Example 36 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} 5 & -3 & 1 & -5 \\ 2 & -1 & 0 & -1 \\ -1 & 0 & 2 & -5 \\ -2 & 2 & -2 & 7 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} 0 & 2 & 1 & 1 \\ -2 & 7 & 2 & 1 \\ 5 & -9 & 1 & 3 \\ 2 & -4 & 0 & 1 \end{array}\right]$

#### Example 37 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 3 & 4 & -6 & 6 \\ 2 & 3 & -5 & 3 \\ -1 & -3 & 8 & 4 \\ -2 & -3 & 8 & 1 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} -27 & 42 & 10 & -4 \\ -11 & 18 & 3 & 0 \\ -12 & 19 & 4 & -1 \\ 9 & -14 & -3 & 1 \end{array}\right]$

#### Example 38 🔗

Show how to find the inverse of the matrix $$C= \left[\begin{array}{cccc} 0 & 0 & 3 & -8 \\ 1 & 0 & 0 & -3 \\ -1 & 2 & -1 & 4 \\ -1 & 1 & 2 & -3 \end{array}\right]$$.

$C^{-1}= \left[\begin{array}{cccc} -15 & 10 & -9 & 18 \\ -4 & 3 & -2 & 5 \\ -13 & 8 & -8 & 16 \\ -5 & 3 & -3 & 6 \end{array}\right]$

#### Example 39 🔗

Show how to find the inverse of the matrix $$C= \left[\begin{array}{cccc} 1 & -2 & 3 & 2 \\ 0 & 1 & -3 & -1 \\ -1 & 2 & -3 & -1 \\ 0 & 1 & -4 & -2 \end{array}\right]$$.

$C^{-1}= \left[\begin{array}{cccc} -2 & 5 & -3 & -3 \\ -2 & 4 & -2 & -3 \\ -1 & 1 & -1 & -1 \\ 1 & 0 & 1 & 0 \end{array}\right]$

#### Example 40 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 1 & 2 & -1 & 6 \\ -1 & -1 & 1 & -7 \\ 0 & 3 & 1 & -8 \\ 0 & 5 & 0 & -4 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 11 & 10 & 1 & -3 \\ -4 & -4 & 0 & 1 \\ -28 & -28 & 1 & 5 \\ -5 & -5 & 0 & 1 \end{array}\right]$

#### Example 41 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 1 & 2 & -6 & 6 \\ 0 & 1 & -3 & 4 \\ 1 & 1 & -2 & 0 \\ 1 & 2 & -4 & 3 \end{array}\right]$$.

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

#### Example 42 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 5 & 5 & -1 & 1 \\ 3 & 4 & -3 & -3 \\ -4 & -3 & -2 & -7 \\ -4 & -4 & 1 & 0 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} -23 & 4 & -5 & -21 \\ 27 & -5 & 6 & 24 \\ 16 & -4 & 4 & 13 \\ -3 & 1 & -1 & -2 \end{array}\right]$

#### Example 43 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} -1 & 0 & 3 & 6 \\ 1 & -1 & -1 & 1 \\ -1 & 2 & 0 & -7 \\ 1 & -2 & -3 & 5 \end{array}\right]$$.

$Q^{-1}= \left[\begin{array}{cccc} 11 & 24 & 15 & 3 \\ 16 & 33 & 22 & 5 \\ -2 & -4 & -3 & -1 \\ 3 & 6 & 4 & 1 \end{array}\right]$

#### Example 44 🔗

Show how to find the inverse of the matrix $$Q= \left[\begin{array}{cccc} 0 & -1 & 0 & -1 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 1 & 5 \\ 1 & 1 & 1 & 7 \end{array}\right]$$.

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

#### Example 45 🔗

Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} 0 & -1 & 0 & 0 \\ 1 & 5 & -1 & 7 \\ 0 & 2 & 1 & -5 \\ 1 & 8 & -1 & 8 \end{array}\right]$$.

$M^{-1}= \left[\begin{array}{cccc} 1 & 3 & 1 & -2 \\ -1 & 0 & 0 & 0 \\ 17 & -5 & 1 & 5 \\ 3 & -1 & 0 & 1 \end{array}\right]$

#### Example 46 🔗

Show how to find the inverse of the matrix $$B= \left[\begin{array}{cccc} 2 & 5 & 8 & -4 \\ 1 & 3 & 3 & -8 \\ -1 & 0 & -2 & -3 \\ -1 & -3 & -5 & 1 \end{array}\right]$$.

$B^{-1}= \left[\begin{array}{cccc} 24 & -11 & 7 & 29 \\ 29 & -14 & 10 & 34 \\ -21 & 10 & -7 & -25 \\ 6 & -3 & 2 & 7 \end{array}\right]$

#### Example 47 🔗

Show how to find the inverse of the matrix $$A= \left[\begin{array}{cccc} 1 & 2 & -1 & -2 \\ -2 & -3 & 1 & 5 \\ 0 & 0 & 1 & -4 \\ 2 & 4 & -1 & -7 \end{array}\right]$$.

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

#### Example 48 🔗

Show how to find the inverse of the matrix $$C= \left[\begin{array}{cccc} 2 & -2 & -5 & 7 \\ 1 & 0 & -3 & 8 \\ -1 & 1 & 3 & -5 \\ 3 & -2 & -5 & 7 \end{array}\right]$$.

$C^{-1}= \left[\begin{array}{cccc} -1 & 0 & 0 & 1 \\ 12 & 4 & 19 & -3 \\ -11 & -3 & -16 & 3 \\ -4 & -1 & -6 & 1 \end{array}\right]$

#### Example 49 🔗

Show how to find the inverse of the matrix $$N= \left[\begin{array}{cccc} 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & -3 \\ 0 & -1 & 0 & 5 \end{array}\right]$$.

$N^{-1}= \left[\begin{array}{cccc} 1 & 5 & 0 & 4 \\ 0 & 5 & 0 & 4 \\ 0 & 3 & 1 & 3 \\ 0 & 1 & 0 & 1 \end{array}\right]$
Show how to find the inverse of the matrix $$M= \left[\begin{array}{cccc} 1 & 3 & 0 & 7 \\ 0 & 1 & -1 & 3 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & -1 \end{array}\right]$$.
$M^{-1}= \left[\begin{array}{cccc} -1 & 3 & 1 & 2 \\ 3 & -8 & -5 & -3 \\ 0 & 0 & 1 & 0 \\ -1 & 3 & 2 & 1 \end{array}\right]$