## G2 - Determinants

#### Example 1 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -2 & 3 & 0 & 5 \\ -6 & 1 & 6 & 1 \\ 3 & 0 & 0 & -1 \\ -5 & 6 & 4 & -3 \end{array}\right] = -632$

#### Example 2 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -1 & 1 & 2 & -1 \\ 4 & -1 & -3 & 6 \\ 0 & 0 & 1 & -2 \\ -3 & -3 & 5 & 1 \end{array}\right] = -78$

#### Example 3 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -2 & 6 & 2 & -5 \\ -1 & -6 & 0 & -3 \\ -5 & 5 & 0 & -3 \\ -4 & 0 & 1 & 3 \end{array}\right] = -655$

#### Example 4 π

Show how to compute the determinant of the matrix

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

.

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

#### Example 5 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 2 & 6 & -5 & 1 \\ 0 & -2 & -4 & 0 \\ 3 & 6 & 2 & -3 \\ 1 & -4 & -1 & 0 \end{array}\right] = -248$

#### Example 6 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 2 & 5 & 0 & -4 \\ 3 & -3 & 6 & 4 \\ 4 & 3 & -5 & -6 \\ -1 & 0 & 0 & -3 \end{array}\right] = -635$

#### Example 7 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -3 & -2 & -4 & 0 \\ 1 & 0 & 3 & 0 \\ -2 & -4 & -1 & 1 \\ 4 & 5 & 5 & 4 \end{array}\right] = -51$

#### Example 8 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 6 & 0 & 1 & -1 \\ 4 & -1 & -6 & 0 \\ -4 & -2 & -5 & 3 \\ 3 & -4 & -3 & 0 \end{array}\right] = 256$

#### Example 9 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -1 & -1 & 4 & 0 \\ 1 & -1 & -4 & 3 \\ 6 & 6 & 0 & 1 \\ -2 & 1 & 1 & 0 \end{array}\right] = 230$

#### Example 10 π

Show how to compute the determinant of the matrix

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

.

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

#### Example 11 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -5 & -6 & -6 & -6 \\ -3 & -1 & 0 & 0 \\ 2 & 2 & 5 & 2 \\ 2 & -5 & 0 & -1 \end{array}\right] = -265$

#### Example 12 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -5 & 5 & -2 & -1 \\ -1 & 0 & 0 & 0 \\ 5 & 4 & 5 & 5 \\ 4 & 3 & 6 & 1 \end{array}\right] = -156$

#### Example 13 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 6 & -1 & -6 & -1 \\ 3 & 6 & 6 & 0 \\ -5 & 2 & -4 & 3 \\ -4 & 1 & 2 & 0 \end{array}\right] = 486$

#### Example 14 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 1 & -3 & -3 & -2 \\ -2 & 2 & 0 & -1 \\ 1 & -4 & 0 & -6 \\ 1 & 2 & 0 & -3 \end{array}\right] = 180$

#### Example 15 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 3 & 2 & 3 & 0 \\ -4 & 2 & -6 & 1 \\ 4 & -4 & 5 & -2 \\ 6 & -1 & 6 & 0 \end{array}\right] = -45$

#### Example 16 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 0 & -3 & -6 & 5 \\ -1 & 1 & -6 & 3 \\ 0 & 2 & 1 & 0 \\ 6 & 2 & 1 & -1 \end{array}\right] = -237$

#### Example 17 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -4 & -6 & 4 & 0 \\ 6 & 5 & 5 & 0 \\ 1 & 0 & 5 & -1 \\ -6 & -2 & 1 & 2 \end{array}\right] = 288$

#### Example 18 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 3 & 1 & 0 & 1 \\ 1 & 2 & 1 & 0 \\ -4 & -1 & -2 & 6 \\ -2 & 2 & 0 & -2 \end{array}\right] = 68$

#### Example 19 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -2 & 0 & -4 & 2 \\ 2 & -1 & -6 & -6 \\ -4 & 0 & -4 & 4 \\ -2 & -2 & 4 & -3 \end{array}\right] = 24$

#### Example 20 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 1 & 1 & -1 & 2 \\ 0 & 0 & -1 & -3 \\ 2 & -6 & -4 & 3 \\ 5 & -5 & 2 & 0 \end{array}\right] = 298$

#### Example 21 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -4 & 3 & 2 & 6 \\ -1 & 0 & -1 & -3 \\ 5 & 0 & 5 & -1 \\ 6 & 0 & 3 & -5 \end{array}\right] = -144$

#### Example 22 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -1 & 6 & 0 & 1 \\ -1 & 0 & -3 & 0 \\ 6 & -1 & -3 & -3 \\ -2 & -3 & -2 & 3 \end{array}\right] = -337$

#### Example 23 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -6 & -4 & 4 & -5 \\ -1 & -6 & -1 & 6 \\ 4 & -2 & 0 & 2 \\ 0 & 0 & 2 & 1 \end{array}\right] = 600$

#### Example 24 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -4 & -3 & 0 & -5 \\ 1 & 1 & -2 & -3 \\ -5 & -6 & 0 & 6 \\ 3 & -6 & 0 & 5 \end{array}\right] = -786$

#### Example 25 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -4 & 5 & 4 & -5 \\ -3 & -4 & 2 & 5 \\ -2 & 5 & 3 & 2 \\ 1 & 0 & 3 & 0 \end{array}\right] = -868$

#### Example 26 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 0 & 0 & 4 & -5 \\ 4 & 3 & 3 & 2 \\ -3 & 0 & -4 & 6 \\ 5 & 1 & 6 & 5 \end{array}\right] = 425$

#### Example 27 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -3 & 6 & -4 & 3 \\ -4 & 6 & 1 & -1 \\ 1 & 0 & -3 & 0 \\ 6 & -3 & -2 & 0 \end{array}\right] = 246$

#### Example 28 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 4 & 2 & -6 & -3 \\ -2 & 3 & -5 & -1 \\ -1 & -6 & 1 & -5 \\ 0 & -3 & 0 & 1 \end{array}\right] = -721$

#### Example 29 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -3 & -4 & 2 & -3 \\ 4 & -3 & 2 & -2 \\ 1 & 3 & 0 & 0 \\ 4 & -3 & 0 & -2 \end{array}\right] = -110$

#### Example 30 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -5 & 2 & -1 & 1 \\ -4 & 1 & 0 & -4 \\ 3 & 2 & 0 & 4 \\ 0 & 5 & -3 & 2 \end{array}\right] = -187$

#### Example 31 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 0 & 2 & 0 & -5 \\ -2 & 2 & -2 & 4 \\ -4 & 2 & 0 & 5 \\ -1 & -3 & 1 & 4 \end{array}\right] = -64$

#### Example 32 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 3 & 4 & 0 & 4 \\ -3 & 3 & 1 & -1 \\ 4 & 0 & 0 & -3 \\ -6 & -4 & 2 & -4 \end{array}\right] = 218$

#### Example 33 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 4 & -6 & -2 & 4 \\ -3 & -3 & -5 & -4 \\ 6 & -3 & -6 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right] = 84$

#### Example 34 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -2 & -1 & 0 & -1 \\ 4 & -1 & 2 & -2 \\ -1 & 0 & 3 & 0 \\ 2 & 6 & -1 & 2 \end{array}\right] = -111$

#### Example 35 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 6 & 1 & -2 & 1 \\ 0 & 3 & 1 & 0 \\ -1 & -6 & -6 & -2 \\ 5 & -6 & 1 & 2 \end{array}\right] = 71$

#### Example 36 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 5 & 4 & 6 & 5 \\ -6 & -6 & 4 & -6 \\ 3 & 2 & 0 & 6 \\ 0 & -2 & 0 & 1 \end{array}\right] = -292$

#### Example 37 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 2 & 6 & 2 & -2 \\ -1 & 5 & 0 & 1 \\ 1 & -1 & 6 & -5 \\ 0 & -2 & 1 & 0 \end{array}\right] = 80$

#### Example 38 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -3 & 4 & 0 & -6 \\ 4 & -3 & 0 & 1 \\ -1 & 3 & 3 & 6 \\ 3 & 2 & 1 & 2 \end{array}\right] = -203$

#### Example 39 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -4 & 0 & 0 & 4 \\ -4 & -2 & -1 & -6 \\ -3 & 1 & 2 & 4 \\ 6 & 0 & 5 & -4 \end{array}\right] = 184$

#### Example 40 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -3 & -5 & -1 & 3 \\ 3 & -3 & 4 & 3 \\ 6 & 0 & -3 & -2 \\ 0 & -2 & -1 & 0 \end{array}\right] = 168$

#### Example 41 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 6 & -3 & -2 & -3 \\ 6 & -5 & -1 & 4 \\ 3 & 4 & 0 & -5 \\ -2 & 4 & 0 & 6 \end{array}\right] = -540$

#### Example 42 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 3 & 0 & 4 & -3 \\ -2 & -6 & -2 & -6 \\ 1 & 0 & 3 & 0 \\ 5 & -6 & 1 & 1 \end{array}\right] = -534$

#### Example 43 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -1 & -3 & 1 & -5 \\ -3 & 6 & -4 & 6 \\ 5 & 3 & -4 & -4 \\ 0 & -1 & -3 & 0 \end{array}\right] = 731$

#### Example 44 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 5 & -1 & 0 & -4 \\ -3 & 2 & 1 & -6 \\ 0 & 5 & 0 & 2 \\ 6 & 5 & 2 & -3 \end{array}\right] = -431$

#### Example 45 π

Show how to compute the determinant of the matrix

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

.

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

#### Example 46 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -2 & 3 & -3 & 1 \\ 6 & 4 & 4 & 0 \\ 4 & -1 & 0 & 0 \\ -4 & -4 & -3 & 3 \end{array}\right] = 332$

#### Example 47 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -5 & 0 & 3 & -2 \\ -5 & -6 & -6 & 4 \\ 1 & -2 & 0 & 0 \\ -6 & 1 & 6 & -2 \end{array}\right] = 216$

#### Example 48 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -5 & -4 & -2 & -3 \\ -5 & -1 & 0 & 1 \\ 0 & -1 & -3 & 0 \\ 2 & 2 & -6 & -3 \end{array}\right] = -307$

#### Example 49 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} 6 & -6 & 1 & 0 \\ -4 & -2 & 2 & 0 \\ -6 & 4 & -1 & -1 \\ -3 & 4 & 1 & 0 \end{array}\right] = -70$

#### Example 50 π

Show how to compute the determinant of the matrix

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

.

$\operatorname{det}\ \left[\begin{array}{cccc} -6 & -2 & -1 & 3 \\ -2 & 0 & 0 & -4 \\ 1 & 1 & 3 & -5 \\ -4 & 0 & 1 & 3 \end{array}\right] = -108$