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Chapter 1: Q34 E (page 1)

If A2 is invertible, then matrix A itself must be invertible.

Short Answer

Expert verified

The statement is true.

Step by step solution

01

Explaining

It is given that is invertible. Need to verify whether A is invertible or not. Suppose that A is not invertible. If matrix A is not invertible, then the matrix is singular. That is, det A = 0.

Now,

detA2=detA.A=detA路detA=0路0=0

02

Result

This contradicts the given statement thatA2 is invertible because an invertible matrix determinant is non-zero but it is found that detA2=0. Hence, our supposition that A is not invertible is not true. So, A is invertible. Therefore, it is true that if is invertible, then A is invertible.

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