Answer to Question #198537 in Linear Algebra for prince

Question

Evaluate det(−A) and det(−A^T). Compare det(−A) and det(−A^T) for:

(2.1) A = −4 2

3 −3 ;

(2.2) A = 3 1 −2

−5 3 −6

−1 0 −4

Answer 1

2.1)

det( A ) = $\begin{vmatrix} -4 & 2 \\ 3 & -3 \\ \end{vmatrix}$

∴ det( -A ) = $\begin{vmatrix} 4 & -2 \\ -3 & 3 \end{vmatrix}$

Expanding along 1^st row, we have

det(-A) = 4 * 3 - (- 3 ) * (- 2 )

det( -A ) = 6

Again, det( - A^T) = $\begin{vmatrix} 4 & -3 \\ -2 & 3 \\ \end{vmatrix}$

Expanding along 1^st row, we have

det(-A) = 4 * 3 - (- 2 ) * (- 3 )

det(-A^T) = 6

Hence, we see that det( -A ) = det(-A^T) .

2.2)

det(A) = $\begin{vmatrix} 3 & 1 & -2 \\ -5 & 3 & -6 \\ -1&0&-4\\ \end{vmatrix}$

So, det( - A ) = $\begin{vmatrix} -3 & -1 & 2 \\ 5 & -3 & 6 \\ 1&0&4\\ \end{vmatrix}$

Expanding along 1^st row, we have

det(-A) = -3[ ( -3 ) $*$ 4 - 0 $*$ 6] - ( -1 )[5 $*$ 4 - 1 $*$ 6] + 2[ 5 $*$ 0 - 1 $*$ (-3)]

det(-A) = 56

det(- A^T) = $\begin{vmatrix} -3 & 5 & 1 \\ -1 & -3 & 0 \\ 2&6&4\\ \end{vmatrix}$

Expanding along 1^st row, we have

det(-A) = -3[ ( -3 ) $*$ 4 - 6 $*$ 0] - 5 [ ( -1 ) $*$ 4 - 2 $*$ 0] + 1[ ( -1 ) $*$ 6 - 2 $*$ ( -3 ) ]

det(- A^T) = 56

Hence, we see that det( -A ) = det(-A^T) .