Let $\mathrm{A}=\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)$. Then the number of $3 \times 3$ matrices B with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is ____

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Matrices

Let

A = 010100001

. Then the number of

3 \times 3

matrices B with entries from the set

{1, 2, 3, 4, 5}

and satisfying

A B = B A

is ____

Answer: 3125

A

is the identity with its first two rows interchanged, so

A B

B

with rows

1

and

2

swapped while

B A

B

with columns

1

and

2

swapped. Writing

B = [b_{ij}]

A B = b_{21} b_{11} b_{31} b_{22} b_{12} b_{32} b_{23} b_{13} b_{33}, B A = b_{12} b_{22} b_{32} b_{11} b_{21} b_{31} b_{13} b_{23} b_{33}

Matching entries gives

b_{11} = b_{22}

b_{12} = b_{21}

b_{13} = b_{23}

and

b_{31} = b_{32}

, while

b_{33}

is unrestricted. So

b_{11}, b_{12}, b_{13}, b_{31}, b_{33}

can be chosen freely from

{1, 2, 3, 4, 5}

and every other entry is then fixed. The number of such matrices is

5^{5} = 3125

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Integer answer