Given tre following matrices, if possible, determine $\mathrm{A}^{2}$. Identify the dimensions of the resulting matrix and fill out the matrix, if it exists. If ${ }^{2}$ not, state "Not Possible".
\[
A=\left[\begin{array}{cc}
8 & -10 \\
-4 & -8
\end{array}\right] \quad B=\left[\begin{array}{c}
-2 \\
-9 \\
-6
\end{array}\right]
\]

Answer

Selecting the check box will replace the entered answer value(s) with the check box value. Entering the dimensions of a matrix will display any boxes necessary to crea matrix.
\[
\square \times \square \square \text { Not Possible }
\]

Step 1 ：We are given the matrix \(A = \left[\begin{array}{cc} 8 & -10 \ -4 & -8 \end{array}\right]\) and we are asked to find \(A^2\).

Step 2 ：The formula for matrix multiplication is given by \((A*B)_{ij} = \sum_{k=1}^{n} (A_{ik} * B_{kj})\), where \(A\) and \(B\) are matrices, \(i\) and \(j\) are the row and column indices of the resulting matrix, and the sum is over \(k\), which ranges over the number of columns in \(A\) (or equivalently, the number of rows in \(B\)).

Step 3 ：In this case, since we are squaring matrix \(A\), both \(A\) and \(B\) in the formula are the same matrix, and \(n\) is 2. We can calculate each element of the resulting matrix using this formula.

Step 4 ：By applying the formula, we find that \(A^2 = \left[\begin{array}{cc} 104 & 0 \ 0 & 104 \end{array}\right]\).

Step 5 ：The square of the matrix \(A\) is a 2x2 matrix given by \(\boxed{\begin{array}{cc} 104 & 0 \ 0 & 104 \end{array}}\)