Question
If $\text{adj A}=\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \text{and B}=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix},$ find adj AB.
✓
Answer
If A and B are non-singular square matrices of the same order, then adj (AB) = (adj B)(adj A) Now, ATQ$\text{adj A}=\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$
$\text{adj B}=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix}$
So, adj(AB) = (adj B)(adj A)$=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix}\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$
$=\begin{bmatrix} -6 & 5 \\ -2 & -10 \end{bmatrix}$
Hence, $\text{adj (AB)}=\begin{bmatrix} -6 & 5 \\ -2 & -10 \end{bmatrix}$
$\text{adj B}=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix}$
So, adj(AB) = (adj B)(adj A)$=\begin{bmatrix} 1 & -2 \\ -3 & 1 \end{bmatrix}\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix}$
$=\begin{bmatrix} -6 & 5 \\ -2 & -10 \end{bmatrix}$
Hence, $\text{adj (AB)}=\begin{bmatrix} -6 & 5 \\ -2 & -10 \end{bmatrix}$
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