If A is a non-diagonal involutory matrix, then Options: A . &nbspA - I = 0 B . &nbspA + I = 0 C . &nbspA - I is non zero singular D . &nbspnone of these

Answer: Option C : C A 2 = I ⇒ A 2 − I = 0 ⇒ (A+I)(A-I)=0 ∴ either | A + I | = 0 or | A − I | = 0 If | A − I | ≠ 0 , then ( A + I ) ( A − I ) = 0 ⇒ A + I = 0 which is not so ∴ | A − I | and A − I ≠ 0 .

If A is a non-diagonal involutory matrix, then

Question

If A is a non-diagonal involutory matrix, then

Options:

A . A - I = 0

B . A + I = 0

C . A - I is non zero singular

D . none of these

Answer: Option C
:
C

A^{2} = I \Rightarrow A^{2} - I = 0

\Rightarrow

(A+I)(A-I)=0

∴

either

| A + I | = 0

| A - I | = 0

| A - I | \neq 0

, then

(A + I) (A - I) = 0 \Rightarrow A + I = 0

which is not so

∴ | A - I |

and

A - I \neq 0

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