# NCERT Class 12-Mathematics: Exemplar Chapter – 3 Matrices Part 13 (For CBSE, ICSE, IAS, NET, NRA 2022)

Doorsteptutor material for IMO-Level-2 is prepared by world's top subject experts: Get full length tests using official NTA interface: all topics with exact weightage, real exam experience, detailed analytics, comparison and rankings, & questions with full solutions.

**Question 33**:

If , then show that

**Answer**:

As

By matrix multiplication

As we know that:

… Hence proved

**Question 34**:

If and , then show that .

**Answer**:

We have, and

And

Also,

And

Now, [using Eq. (i) ]

Hence Proved.

**Question 35**:

Verify that when

**Answer**:

We have

Hence Proved

**Question 36**:

Prove by Mathematical Induction that , where for any square matrix A.

**Answer**:

By principle of mathematical induction, we say that if a statement is true for and if we assume to be true for some random natural number k and usnig it if we prove to be true we can say that is true for all-natural numbers.

We are given to prove that .

Let be the statement: .

Clearly,

is true

Let be true.

Let՚s take now:

We know that by properties of transpose of a matrix:

Thus,

is true.

Hence,

We can say that: is true for all .