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Q.
For any 2 $\times$ 2 matrix A, if A (adj. A) = $\begin{bmatrix}10&0\\ 0&10\end{bmatrix}$ en | A | is equal to :
Determinants
Solution:
Let A be any 2 × 2 matrix.
Given A (adj A) = $\begin{bmatrix}10&0\\ 0&10\end{bmatrix}$
$\Rightarrow $ A (adj A) = 10 $\begin{bmatrix}1 &0\\ 0&1 \end{bmatrix}$ = 10I ....(i)
where I = identity matrix of order 2 $\times$ 2.
We know $A^{-1} = \frac{1}{|A|}$ (Adj.A)
Pre multiplied by 'A', we get
$AA^{-1} = \frac{A}{|A|}. $ .(Adj A)
$\Rightarrow \ I = \frac{A.Adj (A)}{|A|}$
$\Rightarrow \ A (adj A) = | A | I $ ...(ii)
$\therefore $ From equation (i) and (ii), we have | A | = 10