Question 1

# If A and B are square matrices of size n × n such that A2 − B2 = (A − B) (A + B), then which of the following will be always true? A = B AB = BA either of A or B is a zero matrix either of A or B is an identity matrix

Solution

B.

AB = BA

A2 − B2 = (A − B) (A + B)
A2 − B2 = A2 + AB − BA − B2
⇒ AB = BA

Question 2

## If A2 – A + I = 0, then the inverse of A is A + I A A – I I – A

Solution

D.

I – A

Given A2 – A + I = 0
A–1A2 – A–1A + A–1 – I = A–1⋅0 (Multiplying A–1 on both sides)
⇒ A - I + A-1 = 0 or A–1 = I – A.

Question 3

Solution

C.

2

Question 4

## If P =  is the adjoint of a 3 x3 matrix A and |A| = 4, then α is equal to  4 11 5 0

Solution

B.

11

Given,

|P| = 1(12-12)-α (4-6) +3(4-6)
= 2α -6.
∴ |P| = |adj A | = |A|3-1 = |A|2 = 16
[∵ |adj A| = |A|n-1 order is 3 x3
∴ 2α -6 = 16
2α = 22
α = 11

Question 5

## If S is the set of distinct values of 'b' for which the following system of linear equationsx + y + z = 1x + ay + z = 1ax + by + z = 0has no solution, then S is a singleton an empty set an infinite set a finite set containing two or more elements

Solution

A.

a singleton

but at a = 1 and b =1
First two equations are x +y+ z =1
and third equations is x + y +z = 0
⇒ There is no solution
therefore, b = {1}
⇒ it is a singleton set