Chapter 7.7, Problem 23E

a.

To determine

To find the determinant of a matrix and to expand by cofactors on second row.

Answer to Problem 23E

The determinant of a matrix when expanded by cofactors on second row is,

  $\left|A\right|=170$

Explanation of Solution

Given information:

The matrix is given as,

  $\left[\begin{array}{cccc}6& 0& -3& 5\\ 4& 13& 6& -8\\ -1& 0& 7& 4\\ 8& 6& 0& 2\end{array}\right]$

Calculation:

The value of cofactors of the second row can be obtained as,

  $C_{21}=\left|\begin{array}{ccc}0& -3& 5\\ 0& 7& 4\\ 6& 0& 2\end{array}\right|=282$

Similarly, the other cofactors are,

  $C_{22}=-298C_{23}=174C_{24}=-234$

According to the definition of a determinant, second row expansion must be,

  $\begin{array}{l}\left|A\right|=a_{21}{C}_{21}+a_{22}{C}_{22}+a_{23}{C}_{23}+a_{24}{C}_{24}\\ \left|A\right|=4\left(282\right)+13\left(-298\right)+6\left(174\right)-8\left(-234\right)\\ \left|A\right|=170\end{array}$

Therefore, The determinant of a matrix when expanded by cofactors on second row is,

  $\left|A\right|=170$

b.

To determine

To find the determinant of a matrix and to expand by cofactors on second column.

Answer to Problem 23E

The determinant of a matrix when expanded by cofactors on second column is,

  $\left|A\right|=170$

Explanation of Solution

Given information:

The matrix is given as,

  $\left[\begin{array}{cccc}6& 0& -3& 5\\ 4& 13& 6& -8\\ -1& 0& 7& 4\\ 8& 6& 0& 2\end{array}\right]$

Calculation:

The value of cofactors of the second column can be obtained as,

  $C_{12}=\left|\begin{array}{ccc}4& 6& -8\\ -1& 7& 4\\ 8& 0& 2\end{array}\right|=-708$

Similarly, the other cofactors are,

  $C_{22}=-298C_{32}=-48C_{42}=674$

According to the definition of a determinant, second column expansion must be,

  $\begin{array}{l}\left|A\right|=a_{12}{C}_{12}+a_{22}{C}_{22}+a_{32}{C}_{32}+a_{42}{C}_{42}\\ \left|A\right|=0\left(-708\right)+13\left(-298\right)+0\left(-48\right)+6\left(674\right)\\ \left|A\right|=170\end{array}$

Therefore, The determinant of a matrix when expanded by cofactors on second column is,

  $\left|A\right|=170$