Question

Let $A$ be a square matrix of order $2$ such that $A-I=AA^{T}$ (where $I$ is an identity matrix of order $2$ ), then which one of the following is incorrect statement (Where $\left|\right.A\left|\right.$ represents determinant value of matrix $A$ ).

• A. $\left|A\right|=1$
• B. $adjA=A^{- 1}$
• C. Trace of $A=1$
• D. $A=A^{- 1}$

Answer 1

Answer: (D)

$A-I=AA^{T}...\left(i\right)$
Take transpose both the sides
$A^{T}-I=AA^{T}...\left(ii\right)$
from $\left(\right.i\left.\right)$ and $\left(\right.ii\left.\right)$ $A=A^{T}$
$\Rightarrow A-I=A^{2}$
$\Rightarrow A^{2}-A+I=0$
$\Rightarrow \left|A\right|=1,adj\left(A\right)=A^{- 1}$
Trace of $\left(A\right)=1$