"Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$. Then the number of 3 $\times$ 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________."

Question

Accepted Answer

"Let matrix $$B = \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]$$

$\because$ $AB = BA$

$$\left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]\left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right] = \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]\left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$

$$\left[ {\matrix{ d & e & f \cr a & b & c \cr g & h & i \cr } } \right] = \left[ {\matrix{ b & a & c \cr e & d & f \cr h & g & i \cr } } \right]$$

$\Rightarrow d = b,e = a,f = c,g = h$

$\therefore$ Matrix $$B = \left[ {\matrix{ a & b & c \cr b & a & c \cr g & g & i \cr } } \right]$$

No. of ways of selecting a, b, c, g, i

$= 5 \times 5 \times 5 \times 5 \times 5$

$= {5^5} = 3125$

$\therefore$ No. of matrices B = 3125"

Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$. Then the number of 3 $\times$ 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________.

Solution

About this question

Drill 25 more like these. Every day. Free.

Let $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$. Then the number of 3 $\times$ 3 matrices B with entries from the set {1, 2, 3, 4, 5} and satisfying AB = BA is ____________.

Solution

About this question

More Matrices and Determinants questions

Drill 25 more like these. Every day. Free.