Question

If A is involutory matrix and I is unit matrix of same order, then (I-A)(I+A)
E. Zero matrix
F. A
G. I
H. 2A

Answer 1

Hint: If a matrix multiplied by itself yields the identity matrix, it is said to be an involutory matrix. The matrix that is its own inverse is an involutory matrix. If A\times A = I, the matrix A is referred to as an involutory matrix.

Complete step by step solution: We have given matrix A as an involutory matrix and I is the unit matrix of the same order.
We know that the matrix that is its own inverse is an involutory matrix
A^2 A = I
Therefore, on simplifying we get:
(I-A)(I+A)\\
=I^2-IA+IA-A^2\\
=I-{{A}^{2}} [\because A^2=I]\\
=O
So, (I-A)(I+A) is a zero matrix.

Option ‘A’ is correct


Note: An involutory matrix in mathematics is a square matrix that is its own inverse. In other words, if and only if A^2 = I, where I is the n × n identity matrix, then multiplying by the matrix A results in an involution.

Additional Information Involutory Matrix Properties
A is also an involutory matrix if A and B are involutory matrices with AB = BA.
An involutory matrix is also a block diagonal matrix A derived from an involutory matrix.
Involutory matrices' eigenvalues are always +1 and -1.
Any involutory matrix's determinant is always 1.
Each and every symmetric involutory matrix is an orthogonal involutory matrix, and the reverse is also true.
A matrix A is also involutory for all integers n if A is involutory. If n is even, then An = I, and if n is odd, then An = A.
If and only if A is an identity matrix, an involutory matrix A is an idempotent matrix.