To find the product of the given matrices [tex]\(\left[\begin{array}{ccc}-3 & 3 & 0\end{array}\right]\)[/tex] and [tex]\(\left[\begin{array}{c}-3 \\ 5 \\ -2\end{array}\right]\)[/tex], you need to follow the steps of matrix multiplication.
Matrix multiplication involves taking the rows of the first matrix and the columns of the second matrix and performing the dot product. Here is a step-by-step breakdown:
1. The first matrix is a 1x3 matrix:
[tex]\[
\left[\begin{array}{ccc}-3 & 3 & 0\end{array}\right]
\][/tex]
2. The second matrix is a 3x1 matrix:
[tex]\[
\left[\begin{array}{c}-3 \\ 5 \\ -2\end{array}\right]
\][/tex]
3. To multiply these matrices, multiply each element of the row from the first matrix by the corresponding element of the column from the second matrix and then sum them up. The computation is as follows:
[tex]\[
(-3 \cdot -3) + (3 \cdot 5) + (0 \cdot -2)
\][/tex]
4. Calculate each part separately:
[tex]\[
(-3 \cdot -3) = 9
\][/tex]
[tex]\[
(3 \cdot 5) = 15
\][/tex]
[tex]\[
(0 \cdot -2) = 0
\][/tex]
5. Add these results together:
[tex]\[
9 + 15 + 0 = 24
\][/tex]
Therefore, the result of the matrix multiplication is:
[tex]\[
\left[\begin{array}{c}24\end{array}\right]
\][/tex]
The product of the matrices [tex]\(\left[\begin{array}{ccc}-3 & 3 & 0\end{array}\right]\)[/tex] and [tex]\(\left[\begin{array}{c}-3 \\ 5 \\ -2\end{array}\right]\)[/tex] is [tex]\([24]\)[/tex].
