To determine the product of a [tex]\(3 \times 3\)[/tex] identity matrix and the scalar 3, let's follow a step-by-step process:
1. Recall the structure of a [tex]\(3 \times 3\)[/tex] identity matrix, denoted as [tex]\(I\)[/tex]:
[tex]\[ I = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \][/tex]
2. When multiplying a matrix by a scalar, every element of the matrix is multiplied by that scalar. Here, our scalar is 3.
3. Multiplying the identity matrix [tex]\(I\)[/tex] by 3 results in:
[tex]\[
\left[\begin{array}{ccc}
1 \cdot 3 & 0 \cdot 3 & 0 \cdot 3 \\
0 \cdot 3 & 1 \cdot 3 & 0 \cdot 3 \\
0 \cdot 3 & 0 \cdot 3 & 1 \cdot 3
\end{array}\right]
=
\left[\begin{array}{ccc}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{array}\right]
\][/tex]
4. Examining the given options:
A. [tex]\(\left[\begin{array}{lll}3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3\end{array}\right]\)[/tex]
B. [tex]\(\left[\begin{array}{ccc}3 & 8 & -4 \\ 1 & 3 & 2 \\ -1 & -2 & 3\end{array}\right]\)[/tex]
C. [tex]\(\left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]\)[/tex]
D. [tex]\(\left[\begin{array}{lll}3 & 1 & 1 \\ 1 & 3 & 1 \\ 1 & 1 & 3\end{array}\right]\)[/tex]
5. From these options, it’s clear that option C matches the resulting matrix we calculated.
Therefore, the correct answer is C:
[tex]\[
\left[\begin{array}{ccc}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]
\][/tex]
