Let A be a square matrix of order 2 such that A-I=AAT (where I is an identity matrix of order 2 ), then which one of the following is incorrect statement (Where |.A|. represents determinant value of matrix A ).

Question

Let A be a square matrix of order 2 such that A-I=AAT (where I is an identity matrix of order 2 ), then which one of the following is incorrect statement (Where |.A|. represents determinant value of matrix A ). (A) |A|=1 (B) adjA=A- 1 (C) Trace of A=1 (D) A=A- 1

Tardigrade Admin · Accepted Answer

A-I=AAT...(i) Take transpose both the sides AT-I=AAT...(ii) from (.i.) and (.ii.) A=AT ⇒ A-I=A2 ⇒ A2-A+I=0 ⇒ |A|=1,adj(A)=A- 1 Trace of (A)=1

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

$∣ A ∣ = 1$

$a d j A = A^{- 1}$

Trace of $A = 1$

$A = A^{- 1}$

$A - I = A A^{T} ... (i)$
Take transpose both the sides
$A^{T} - I = A A^{T} ... (ii)$
from $(i)$ and $(ii)$ $A = A^{T}$
$\Rightarrow A - I = A^{2}$
$\Rightarrow A^{2} - A + I = 0$
$\Rightarrow ∣ A ∣ = 1, a d j (A) = A^{- 1}$
Trace of $(A) = 1$

Q. Let A be a square matrix of order 2 such that A−I=AAT (where I is an identity matrix of order 2 ), then which one of the following is incorrect statement (Where ∣A∣ represents determinant value of matrix A ).

∣A∣=1

adjA=A−1

Trace of A=1

A=A−1