Question

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

• A. 0
• B. 10
• C. 20
• D. 100

Answer 1

Answer: (B)

Let A be any 2 × 2 matrix.
Given A (adj A) = $\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$
$\Rightarrow$ A (adj A) = 10 $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ = 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
$A{A}^{-1}=\frac{A}{|A|}.$ .(Adj A)
$\Rightarrow I=\frac{A.Adj(A)}{|A|}$
$\Rightarrow A(adjA)=|A|I$ ...(ii)
$\therefore$ From equation (i) and (ii), we have | A | = 10