Question

For any 2 $\times$ 2 matrix A, if A (adj. A) = $\begin{bmatrix}10&0\\\ 0&10\end{bmatrix}$ en | A | is equal to :

• A. 0
• B. 10
• C. 20
• D. 100

Answer 1

Answer: (B)

10

Answer 2

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