NCERT Class 12-Mathematics: Exemplar Chapter – 3 Matrices Part 17 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Question 48:

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew symmetric.

Answer:

A matrix is said to be skew-symmetric if

Given, B is a skew-symmetric matrix.

Let

We have to prove C is skew-symmetric.

To prove:

As

We know that:

From equation 1 and 2:

We have,

Thus we say that is a skew-symmetric matrix.

Long Answer (L. A)

Question 49:

If for any two sqaure matrices, prove by mathematical induction that .

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:

NOTE: As we know that Matrix multiplication is not commutative. So we can՚t write directly that

But we are given that

As

We observe that one power of B is decreasing while other is increasing. After certain repetitions decreasing power of B will become I

And at last step:

Thus is true when is true.

when .