Say true or false:
Let A, B be two matrices such that they commute, then $(A B)^{n} = A^{n} B^{n}$ .

True

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False

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Solution

The correct option is A True
$A$ and $B$ commute each other then $A B = B A$
${(A B)}^{n} = {(B A)}^{n} \Rightarrow {(A B)}^{n} = B^{n} A^{n} = A^{n} B^{n} = {(B A)}^{n}$

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Similar questions

Q. Let A, B be two matrices such that they commute, then for any positive integer n,
(i)

A B^{n} = B^{n} A

(ii)

(A B)^{n} = A^{n} B^{n}

Q. Say true or false:

A

B

be two matrices such that they commute, then

A^{2} - B^{2} = (A - B) (A + B)

Say true or false:

Let A, B be two matrices, such that

A B = A

and

B A = B

, then

A^{2} = A

and

B^{2} = B

Q. Let

A

and

B

be any two square matrices such that

A B = B A

, then by mathematical induction

(A B)^{n} = A^{n} B^{n}

is true for

If A and B are square matrices of the same order such that AB=BA, then prove by induction that A $B^{n}$ = $B^{n}$ A. Further, prove that $(A B)^{n}$ = $A^{n}$ $B^{n}$ for all $n \in N$ .

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Say true or false:Let A, B be two matrices such that they commute, then (AB)n=AnBn.

The correct option is A TrueA and B commute each other then AB=BA(AB)n=(BA)n⇒(AB)n=BnAn=AnBn=(BA)n

Say true or false:
Let A, B be two matrices such that they commute, then $(A B)^{n} = A^{n} B^{n}$ .

The correct option is A True
$A$ and $B$ commute each other then $A B = B A$
${(A B)}^{n} = {(B A)}^{n} \Rightarrow {(A B)}^{n} = B^{n} A^{n} = A^{n} B^{n} = {(B A)}^{n}$