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|$

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.

As mentioned in the question, we have to find the value of ${{A}^{12}}$ if A $=\left| \left( \begin{matrix} 0 & 0 \\ 0 & 5 \\ \end{matrix} \right) \right|$ .
\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}
$A=0=\left( \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right)$
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
${{A}^{12}}=\left| \left( \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right) \right|$
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.