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Chapter 3: Q21E (page 164)

If A and B are invertible matrices, then AB must be similar to BA.

Short Answer

Expert verified

The above statement is false.

If A and B are two invertible matrices, then AB may not be similar to BA.

Step by step solution

01

Definition of similar matrix

Let A and B are two square matrices, the matrix A is said to be similar to matrix B if there exists an invertible matrix P such that

B=P-1AP

02

Mentioning concept

Since it is given that matrix A and matrix B are invertible, thenA-1andB-1 exist and are also invertible.

If we multiply the matrix AB by A-1from left and from right, we get the matrix

A-1(AB)A=(A-1A)(BA)鈬�A-1(AB)A=BA(鈭�A-1A=I)

This shows that matrix BA is similar to matrix AB.

Now, if we multiply the matrix BA by B-1from left and from right, we get the matrix

B-1(BA)B=(B-1B)(AB)鈬�B-1(BA)B=AB(鈭�B-1B=I)

This shows that matrix AB is similar to matrix BA.

03

Final Answer

If A is an invertible matrix then BA is similar to AB and if B is an invertible matrix then AB is similar to BA.

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