Chapter 8.4, Problem 44E

(a)

To determine

Determinant of the matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$ expanding by cofactors of Row 1

Answer to Problem 44E

Determinant $=64$

Explanation of Solution

Given:

The matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$

Calculation:

The determinant of the matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$ is given by $\left|\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right|$

Expanding the determinant by cofactors of Row 1,

  $\left|\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right|$

  $\begin{array}{l}=7\times \left|\begin{array}{ccc}0& 1& -2\\ -2& 3& 2\\ 0& -1& 4\end{array}\right|-0\times \left|\begin{array}{ccc}6& 1& -2\\ 1& 3& 2\\ -3& -1& 4\end{array}\right|+0\times \left|\begin{array}{ccc}6& 0& -2\\ 1& -2& 2\\ -3& 0& 4\end{array}\right|-(-6)\times \left|\begin{array}{ccc}6& 0& 1\\ 1& -2& 3\\ -3& 0& -1\end{array}\right|\\ =7\times (0-1\times -8-2\times 2)-0+0+6\times (6\times 2-0+1\times -6)\end{array}$

  $\begin{array}{l}=7\times (0+8-4)-0+0+6\times (12-0-6)\\ =7\times 4-0+0+6\times 6\\ =28+36\\ =64\end{array}$

Thus, the determinant of the matrix is 64.

(b)

To determine

Determinant of the matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$ expanding by cofactors of Column2

Answer to Problem 44E

Determinant $=64$

Explanation of Solution

Given:

The matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$

Calculation:

The determinant of the matrix $\left[\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right]$ is given by $\left|\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right|$

Expanding the determinant by cofactors of Column 2,

  $\left|\begin{array}{cccc}7& 0& 0& -6\\ 6& 0& 1& -2\\ 1& -2& 3& 2\\ -3& 0& -1& 4\end{array}\right|$

  $\begin{array}{l}=-0\times \left|\begin{array}{ccc}6& 1& -2\\ 1& 3& 2\\ -3& -1& 4\end{array}\right|+0\times \left|\begin{array}{ccc}7& 0& -6\\ 1& 3& 2\\ -3& -1& 4\end{array}\right|-(-2)\times \left|\begin{array}{ccc}7& 0& -6\\ 6& 1& -2\\ -3& -1& 4\end{array}\right|+0\times \left|\begin{array}{ccc}7& 0& -6\\ 6& 1& -2\\ 1& 3& 2\end{array}\right|\\ =0+0+2\times (7\times 2-0-6\times -3)+0\\ =0+0+2\times 32+0\\ =64\end{array}$

Thus, the determinant of the matrix is 64.