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Question

Mathematics Question on Matrices

If A=[333 333 333 ],A= \begin{bmatrix} 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ \end{bmatrix} , then A4{{A}^{4}} is equal to

A

27 A27 \,A

B

81 A81\, A

C

243 A243\, A

D

729 A729\, A

Answer

729 A729\, A

Explanation

Solution

Given, A=[333 333 333 ]A=\left[ \begin{matrix} 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ \end{matrix} \right]
∴\therefore A=3∣111 111 111 ∣A=3\left| \begin{matrix} 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ \end{matrix} \right|
∴\therefore A2=3[111 111 111 ].3[111 111 111 ]{{A}^{2}}=3\left[ \begin{matrix} 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ \end{matrix} \right].3\left[ \begin{matrix} 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ \end{matrix} \right]
=9[333 333 333 ]=9A=9\left[ \begin{matrix} 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ 3 & 3 & 3 \\\ \end{matrix} \right]=9A
∴\therefore A4=A2.A2{{A}^{4}}={{A}^{2}}.{{A}^{2}}
=9A.9A=81A2=81.9A=9A.9A=81{{A}^{2}}=81.9A
=729A=729A