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

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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 .