If for the non-singular matrix A, A2 = I, then find A-1.

Question

If for the non-singular matrix A, A2 = I, then find A-1. (A) A (B) I (C) O (D) None of these

Tardigrade Admin · Accepted Answer

Given, A2 = I Since, A is non-singular matrix ∴ |A|≠0, so, A-1 exists. Multiplying by A-1 on both sides, we get A-1(A2)=A-1(I) ⇒ (A-1A)A=A-1 ⇒ IA=A-1 (∵ A-1A=I) ∴ A-1=A (∵ IA=A)

Q. If for the non-singular matrix $A, A^{2} = I$ , then find $A^{- 1}$ .

$A$

$I$

$O$

None of these

Given, $A^{2} = I$
Since, A is non-singular matrix
$∴ ∣ A ∣ \neq = 0, so, A^{- 1}$ exists.
Multiplying by $A^{- 1}$ on both sides, we get
$A^{- 1} (A^{2}) = A^{- 1} (I)$
$\Rightarrow (A^{- 1} A) A = A^{- 1} \Rightarrow I A = A^{- 1} (∵ A^{- 1} A = I)$
$∴ A^{- 1} = A (∵ I A = A)$

Q. If for the non-singular matrix A,A2=I, then find A−1.

A

I

O

None of these

Given, A2=I Since, A is non-singular matrix ∴∣A∣=0,so,A−1 exists. Multiplying by A−1 on both sides, we get A−1(A2)=A−1(I) ⇒(A−1A)A=A−1⇒IA=A−1(∵A−1A=I) ∴A−1=A(∵IA=A)

Q. If for the non-singular matrix $A, A^{2} = I$ , then find $A^{- 1}$ .

$A$

$I$

$O$

Given, $A^{2} = I$
Since, A is non-singular matrix
$∴ ∣ A ∣ \neq = 0, so, A^{- 1}$ exists.
Multiplying by $A^{- 1}$ on both sides, we get
$A^{- 1} (A^{2}) = A^{- 1} (I)$
$\Rightarrow (A^{- 1} A) A = A^{- 1} \Rightarrow I A = A^{- 1} (∵ A^{- 1} A = I)$
$∴ A^{- 1} = A (∵ I A = A)$