Chapter 7.7, Problem 21E

a.

To determine

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

Answer to Problem 21E

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

  $\left|A\right|=-75$

Explanation of Solution

Given information:

The matrix is given as,

  $\left[\begin{array}{ccc}-3& 2& 1\\ 4& 5& 6\\ 2& -3& 1\end{array}\right]$

Calculation:

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

  $C_{11}=\left|\begin{array}{cc}5& 6\\ -3& 1\end{array}\right|=23$

Similarly, the other cofactors are,

  $C_{12}=8C_{13}=-22$

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

  $\begin{array}{l}\left|A\right|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}\\ \left|A\right|=-3\left(23\right)+2\left(8\right)+1\left(-22\right)\\ \left|A\right|=-75\end{array}$

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

  $\left|A\right|=-75$

b.

To determine

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

Answer to Problem 21E

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

  $\left|A\right|=-75$

Explanation of Solution

Given information:

The matrix is given as,

  $\left[\begin{array}{ccc}-3& 2& 1\\ 4& 5& 6\\ 2& -3& 1\end{array}\right]$

Calculation:

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

  $C_{12}=\left|\begin{array}{cc}4& 6\\ 2& 1\end{array}\right|=8$

Similarly, the other cofactors are,

  $C_{22}=-5C_{32}=22$

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

  $\begin{array}{l}\left|A\right|=a_{12}{C}_{12}+a_{22}{C}_{22}+a_{32}{C}_{32}\\ \left|A\right|=2\left(8\right)+5\left(-5\right)-3\left(22\right)\\ \left|A\right|=-75\end{array}$

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

  $\left|A\right|=-75$