Question-9

Suppose \(A=\begin{bmatrix}4 & -1 & 3\\ 2 & 0 & 1\\ 3 & -2 & 0 \end{bmatrix}\). Find the determinant of the cofactor matrix of \(A\).

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We have the following relation:

\[ A \cdot \text{adj}(A)=\text{adj}(A)\cdot A=\text{det}(A)\cdot I \]

We also know that \(\text{adj}(A)=C^{T}\), where \(C\) is the cofactor matrix. Therefore, \(|\text{adj}(A)|=\text{det}|C^{T}|=\text{det}|C|\). It is enough if we compute the determinant of the adjugate:

\[ \begin{aligned} A\cdot\text{adj}(A) & =\text{det}(A)\cdot I\\ \implies\text{det}(A)\cdot\text{det}(\text{adj}(A)) & =\text{det}(A)^{3}\\ \implies\text{det}(\text{adj}(A)) & =\text{det}(A)^{2} & \text{det}(A)\neq0 \end{aligned} \]

Let us now compute \(\text{det}(A)\):

\[ \begin{aligned} \begin{vmatrix}4 & -1 & 3\\ 2 & 0 & 1\\ 3 & -2 & 0 \end{vmatrix} & =4\times(0+2)+1(0-3)+3(-4-0)\\ & =8-3-12\\ & =-7 \end{aligned} \]

So the determinant of the cofactor matrix is \(49\).