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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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Most popular questions from this chapter

Question: Are the columns of an invertible matrix linearly independent?

In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

17. [0120300014]

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

54.(0,0)(1,0)(2,0)(0,1)(1,1)(2,1)(0,2)(1,2)(2,2).

Find a basis of the image of the matrix [012030000140000001000000].

In Exercises 37 through 42 , find a basisJ of 鈩�n such that theJ-matrix of the given linear transformation T is diagonal.

Orthogonal projection T onto the line in鈩�3 spanned by[123].
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