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 .