You visited us 1 times! Enjoying our articles? Unlock Full Access!

Addition and Subtraction of a Matrix

Let A, B be t...

Question

Let A, B be two matrices such that they commute, then for any positive integer n,
(i) $A B^{n} = B^{n} A$ (ii) $(A B)^{n} = A^{n} B^{n}$

only

(i)

is correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

both

(i)

and

(i i)

are correct

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

only

(i i)

is correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

none of

(i)

and

(i i)

is correct

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A both $(i)$ and $(i i)$ are correct
Given, $A B = B A$
Now, for (i)
Taking $A B^{2} = (A B) B$
$= (B A) B$
$= B (A B)$
$= B (B A) = B^{2} A$
$\Rightarrow (A B^{2}) = B^{2} A$
$A B^{3} = (A B^{2}) B = (B^{2} A) B$
$= B^{2} (A B) = B^{2} B A$
$= B^{3} A$
$\Rightarrow A B^{3} = B^{3} A$
Hence, $A B^{n} = B^{n} A$
Now, for (ii)
Taking $(A B)^{2} = (A B) (A B)$
$= A (B A) B$
$= A (A B) B = A^{2} B^{2}$
$\Rightarrow (A B)^{2} = A^{2} B^{2}$
Now, $(A B)^{3} = (A B) (A B) (A B)$
$= A (B A) (B A) B$
$= A (A B) (A B) B$
$= A^{2} (B A) B^{2}$
$= A^{2} (A B) B^{2}$
$= A^{3} B^{3}$
Hence, for $(A B)^{n} = A^{n} B^{n}$

Suggest Corrections

Similar questions

Q. Say true or false:

Let A, B be two matrices such that they commute, then

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

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 $n \in N$ .

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

Q. Let

x_{k}

be real numbers such that

x_{k} \geq k^{4} + k^{2} + 1

for

1 \leq k \leq 2018

. Denote

N = 2018 \sum k = 1 k

. Consider the following inequalities :-
I.

{(2018 \sum k = 1 k x_{k})}_{k}^{2} \leq N (2018 \sum k = 1 k {x_{k}}^{2})

II.

{(2018 \sum k = 1 k x_{k})}_{k}^{2} \leq N (2018 \sum k = 1 k^{2} {x_{k}}^{2})

Q. Consider the quadratic equation

n x^{2} + 7 \sqrt{n} x + n = 0

, where

n

is a positive integer. Which of the following statements are necessarily correct ?

I. For any

n

, the roots are distinct.
II. There are infinitely many values of

n

for which both roots are real.
III. The product of the roots is necessarily an integer

Join BYJU'S Learning Program

Let A, B be two matrices such that they commute, then for any positive integer n,(i) ABn=BnA (ii) (AB)n=AnBn

Let A, B be two matrices such that they commute, then for any positive integer n,
(i) $A B^{n} = B^{n} A$ (ii) $(A B)^{n} = A^{n} B^{n}$