Question

If A is a invertible idempotent matrix of order n, then adj A is equal to

• A. $(adjA{)}^2$
• B. $I$
• C. $A^1$
• D. none of these

Answer 1

Answer: (A), (B), (C)

$I$
$A^1$
$(adjA{)}^2$

Given $A^2=A$ $A^2-A=0$

$\Rightarrow adjA={\left(AdjA\right)}^2$ $\Rightarrow A(A-I)=0$

$A=0$ $A=I$

A is invertible

$\Rightarrow A\ne 0$

$\to A=I$

$\Rightarrow AdjA={\left(AdjA\right)}^2=A=I$

Option: A, B,C are true.