Let A=[a&0&0 0&a&0 0&0&a], then An is equal to

Question

Let A=[a&0&0 0&a&0 0&0&a], then An is equal to (A) [an&0&0 0&an&0 0&0&a]  (B) [an&0&0 0&a&0 0&0&a]  (C) [an&0&0 0&an&0 0&0&an]  (D) [na&0&0 0&na&0 0&0&na]

Tardigrade Admin · Accepted Answer

A2=[a&0&0 0&a&0 0&0&a][a&0&0 0&a&0 0&0&a]=[a2&0&0 0&a2&0 0&0&a2] A3=[a2&0&0 0&a2&0 0&0&a2][a&0&0 0&a&0 0&0&a]=[a3&0&0 0&a3&0 0&0&a3]

Q. Let $A = ⎣ ⎡ a 00 0 a 0 00 a ⎦ ⎤,$ then $A^{n}$ is equal to

$⎣ ⎡ a^{n} 00 0 a^{n} 0 00 a ⎦ ⎤$

$⎣ ⎡ a^{n} 00 0 a 0 00 a ⎦ ⎤$

$⎣ ⎡ a^{n} 00 0 a^{n} 0 00 a^{n} ⎦ ⎤$

$⎣ ⎡ na 00 0 na 0 00 na ⎦ ⎤$

$A^{2} = ⎣ ⎡ a 00 0 a 0 00 a ⎦ ⎤ ⎣ ⎡ a 00 0 a 0 00 a ⎦ ⎤ = ⎣ ⎡ a^{2} 00 0 a^{2} 0 00 a^{2} ⎦ ⎤$
$A^{3} = ⎣ ⎡ a^{2} 00 0 a^{2} 0 00 a^{2} ⎦ ⎤ ⎣ ⎡ a 00 0 a 0 00 a ⎦ ⎤ = ⎣ ⎡ a^{3} 00 0 a^{3} 0 00 a^{3} ⎦ ⎤$

Q. Let A=⎣⎡​a00​0a0​00a​⎦⎤​, then An is equal to

⎣⎡​an00​0an0​00a​⎦⎤​

⎣⎡​an00​0a0​00a​⎦⎤​

⎣⎡​an00​0an0​00an​⎦⎤​

⎣⎡​na00​0na0​00na​⎦⎤​