12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that...

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12. Let A = CD , where C is an invertible n × n matrix and A and D are n × n matrices. Prove that the matrix DC is similar to

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Answer 1

Answer #1

Two matrices M and N are similar if there exist a matrix P such that

$N = P^{-1}MP$

Here:

M = DC

and

N = A

So,

To prove:

We can find matrix P such that:

$A= P^{-1}DCP$ (1)

Since given:

$A= CD$ (2)

Substituting (2) in equation (1), we get:

To prove:

$CD= P^{-1}DCP$ (3)

Choose:

P = D (4)

Substituting (4), equation (3) becomes:

$CD= D^{-1}DCD$

$= I\times CD=CD$

where I is the Identity Matrix.

Thus, we note that by choosing P=D, equation (1) is satisfied.

Thus, we prove that the matrix DC is similar to A.

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Answer 2

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