JEE MAIN - Mathematics (2021 - 22th July Evening Shift - No. 17)
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 ____________.
Answer
3125
Explanation
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
$$\because$$ $$AB = BA$$
$$\left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]\left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right] = \left[ {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right]\left[ {\matrix{ 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 1 \cr } } \right]$$
$$\left[ {\matrix{ d & e & f \cr a & b & c \cr g & h & i \cr } } \right] = \left[ {\matrix{ b & a & c \cr e & d & f \cr h & g & i \cr } } \right]$$
$$ \Rightarrow d = b,e = a,f = c,g = h$$
$$\therefore$$ 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
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