Question

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

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

{{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{aligned} & 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{aligned}$$ Now, as A is equal to 0, hence, A can be written as follows $$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.