Question

If A \[=\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 5 \\
\end{matrix} \right) \right|\] , then \[{{A}^{12}}\] is:
\[\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 60 \\
\end{matrix} \right) \right|\]
 \[\left| \left( \begin{matrix}
   0 & 0 \\
   0 & {{5}^{12}} \\
\end{matrix} \right) \right|\]
\[\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right) \right|\]
\[\left| \left( \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right) \right|\]

Answer 1

Hint: For finding the determinant of order 2 matrix, the formula for that is as follows
\[\left| \left( \begin{matrix}
   {{a}_{11}} & {{a}_{12}} \\
   {{a}_{21}} & {{a}_{22}} \\
\end{matrix} \right) \right|=\left( {{a}_{11}}\cdot {{a}_{22}}-{{a}_{12}}\cdot {{a}_{21}} \right)\]
Another important thing to be known is that a singular matrix is a matrix whose determinant is equal to 0 and also a singular matrix raised to any power produces a singular matrix only.

Complete step-by-step answer:
As mentioned in the question, we have to find the value of \[{{A}^{12}}\] if A \[=\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 5 \\
\end{matrix} \right) \right|\] .
Now, for calculating the value of A which is nothing but the determinant value of a matrix, we will use the formula which is given in the hint that is as follows
\[\begin{align}
  & A=\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 5 \\
\end{matrix} \right) \right| \\
 & A=\left( 0\cdot 5-0\cdot 0 \right) \\
 & A=0 \\
\end{align}\]
Now, as A is equal to 0, hence, A can be written as follows
\[A=0=\left( \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right)\]
Now, as A is a singular matrix, hence, A raised to any power is also a singular matrix. Therefore, the value of \[{{A}^{12}}\] is also a singular matrix that is
\[{{A}^{12}}=\left| \left( \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right) \right|\]
Hence, the correct option is option (c).

Note: The students can make an error in evaluating the value of \[{{A}^{12}}\] if they don’t know the what is a singular matrix or what are the properties of a singular matrix that are given in the hint which is to be known is that a singular matrix is a matrix whose determinant is equal to 0 and also a singular matrix raised to any power produces a singular matrix only.