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Consider the following statements: 1. If A′ = A; then A is a singular matrix, where A′ is the transpose of A. 2. If A is a square matrix such that A3 = I, then A is non-singular. Which of the statements given above is/are correct ?

Consider the following statements: 1. If A′ = A; then A is a singular matrix, where A′ is the transpose of A. 2. If A is a square matrix such that A3 = I, then A is non-singular. Which of the statements given above is/are correct ? Correct Answer 2 only 

Concept:

A matrix is singular if and only if its determinant is zero.

 

Calculation:

If A′ = A; where A′ is the transpose of A then |A| = |A'|

But it is not necessary that |A| = 0, so A is not a  singular matrix.

Hence, Statement 1 is wrong.

 

Given, A3 = I

Taking determinants both sides, we get

⇒|A3 | = |I| = 1

⇒ |A| = 1

Here, |A|≠ 0 so, A is a non-singular matrix

Hence, option (2) is correct.  

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