If A is an Involuntary matrix and I is a unit matrix of same order, then (I − A) (I + A) is

  1. A
  2. I
  3. 2A
  4. Zero matrix

Answer (Detailed Solution Below)

Option 4 : Zero matrix
Free
NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins

Detailed Solution

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Concept:

Involuntary matrix:

  • Matrix A is said to be Involuntary if A2 = I, where I is an Identity matrix of same order as of A.
  • Involuntary matrix is a matrix that is equal to its own inverse. ⇔ A-1 = A

 

Calculation:

Given that A is involuntary matrix,

⇒ A2 = I

Now,

(I − A) (I + A) = I2 – IA + AI − A2 

⇒ I – A + A – I         (∵ A2 = I)

0

∴ (I − A) (I + A) is zero matrix.

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