To find the elements of the resulting matrix [tex]\( A^2 \)[/tex] for the given matrix [tex]\( A \)[/tex], we need to perform the matrix multiplication of [tex]\( A \)[/tex] with itself.
Given the matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
0 & -6 \\
-3 & -2
\end{pmatrix}
\][/tex]
We need to calculate [tex]\( A^2 = A \times A \)[/tex].
Performing the matrix multiplication step-by-step:
1. First row, first column element of [tex]\( A^2 \)[/tex]:
[tex]\[
(0 \times 0) + (-6 \times -3) = 0 + 18 = 18
\][/tex]
2. First row, second column element of [tex]\( A^2 \)[/tex]:
[tex]\[
(0 \times -6) + (-6 \times -2) = 0 + 12 = 12
\][/tex]
3. Second row, first column element of [tex]\( A^2 \)[/tex]:
[tex]\[
(-3 \times 0) + (-2 \times -3) = 0 + 6 = 6
\][/tex]
4. Second row, second column element of [tex]\( A^2 \)[/tex]:
[tex]\[
(-3 \times -6) + (-2 \times -2) = 18 + 4 = 22
\][/tex]
So, the resulting matrix [tex]\( A^2 \)[/tex] is:
[tex]\[
A^2 = \begin{pmatrix}
18 & 12 \\
6 & 22
\end{pmatrix}
\][/tex]
Thus, the elements of the matrix [tex]\( A^2 \)[/tex] are 18, 12, 6, and 22.
