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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

The column vectors of a 5脳4 matrix must be linearly dependent.

Question: Consider an n脳pmatrix Aand ap脳mmatrix B. We are told that the columns of A and the columns of B are linearly independent. Are the columns of the product AB linearly independent as well?

Express the image of the matrix

A=[1116123413521470]

as the kernel of a matrixB. Hint: The image ofAconsists of all vectorsy鈫in鈩4such that the systemAx鈫=y鈫is consistent. Write this system more explicitly:

localid="1664197199135" |x1+鈥鈥鈥鈥x2+鈥鈥鈥鈥x3+6x4=y1x1+2x2+3x3+4x4=鈥y2x1+3x2+5x3+2x4=鈥y3x1+4x2+7x3鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥=鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥y4|.

Now, reduce rows:

x1-鈥鈥鈥鈥鈥鈥鈥鈥x3+8x4=4y3-3y4x2+2x3-2x4=鈥鈥鈥鈥鈥-y3+y40=鈥鈥鈥鈥鈥-3y3+2y4鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥鈥0=鈥鈥y2-鈥2y3+y4

For which the vectors y鈫is this system consistent? The answer allows you to express im ( A) as the kernel of a2脳4matrix B.

Find a basis of the subspace of 鈩3defined by the equation2x1+3x2+x3=0.

Consider an 鈥n脳pmatrix 鈥Aand鈥p脳mmatrix B.

  1. What is the relationship between ker(AB) and ker role="math" localid="1664188035115" B)? Are they always equal? Is one of them always contained in the other?
  2. What is the relationship between im(A) and im (role="math" localid="1664187975738" AB)?
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