If $A=\left[ \left( \begin{matrix}
   3 & 1 \\
   -1 & 2 \\
\end{matrix} \right) \right]$ then ${{A}^{2}}$is equal to ?
A . $\left( \begin{matrix}
   8 & -5 \\
   -5 & 3 \\
\end{matrix} \right)$
B. $\left( \begin{matrix}
   8 & -5 \\
   5 & 3 \\
\end{matrix} \right)$
C. $\left( \begin{matrix}
   8 & -5 \\
   -5 & -3 \\
\end{matrix} \right)$
D. $\left( \begin{matrix}
   8 & 5 \\
   -5 & 3 \\
\end{matrix} \right)$

Hint: We are given a question which is based on matrices. We are given a matrix and we have to find the square of that matrix. A square matrix is a matrix that has the same number of rows and columns. For example – The given matrix is $2\times 2$ matrix. To Find the square of A we multiply A to A and after simplifying it, we get the value of the square of matrix A.

Complete step by step Solution:
Given $A=\left[ \left( \begin{matrix}
   3 & 1 \\
   -1 & 2 \\
\end{matrix} \right) \right]$
We have to find the value of ${{A}^{2}}$
${{A}^{2}}=A\times A$
Now we put the value of A in the above equation and we get,
Then ${{A}^{2}}=\left( \begin{matrix}
   3 & 1 \\
   -1 & 2 \\
\end{matrix} \right)\times \left( \begin{matrix}
   3 & 1 \\
   -1 & 2 \\
\end{matrix} \right)$
Now we open the brackets of R.H.S and multiply the terms, we get
${{A}^{2}}=\left( \begin{matrix}
   3\times 3+1\times -1 & 3\times 1+1\times 2 \\
   -1\times 3+2\times -1 & -1\times 1+2\times 2 \\
\end{matrix} \right)$
Adding the terms and solving it, we get
${{A}^{2}}=\left( \begin{matrix}
   9-1 & 3+2 \\
   -3-2 & -1+4 \\
\end{matrix} \right)$
Simplifying further, we get
${{A}^{2}}=\left( \begin{matrix}
   8 & 5 \\
   -5 & 3 \\
\end{matrix} \right)$
Hence the value of ${{A}^{2}}=\left( \begin{matrix}
   8 & 5 \\
   -5 & 3 \\
\end{matrix} \right)$

Therefore, the correct option is (D).

Note: We know that the given question is in matrix form. A matrix is a set of numbers that are arranged in rows and columns to make a rectangular array. In the matrix, the numbers are called the entries or entities of the matrix.
In Multiplication matrices, the number of column of the first matrix match the number of rows of the second matrix. When we want to multiply the matrices, then the parts of the rows in the first matrix are multiplied by the columns in the second matrix.