Maths-

General

Easy

Question

# Assertion : There are only finitely many 2 × 2 matrices which commute with the matrix

Reason : If A is non-singular then it commutes with I, adj A and A^{–1}

- If both (A) and (R) are true, and (R) is the correct explanation of (A).
- If both (A) and (R) are true but (R) is not the correct explanation of (A).
- If (A) is true but (R) is false.
- If (A) is false but (R) is true.

## The correct answer is: If (A) is false but (R) is true.

### The reason R is true since

AI = IA, AA^{–1} = A^{–1}A = I, A|adj A| = |adj. A|A

But a matrix can commute with general order matrices which may be infinite in number.

Let B = be a matrix which commute with A then AB = BA

The above four relations are equivalent to only two independent relations

a – d = b, b + 2c = 0

If d = , then a = b + = –2c +

Thus, are all possible 2 × 2 matrices which commute with given matrix A =

and c being any arbitrary complex numbers. Thus assertion is therefore false.

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