To solve the matrix multiplication problem:
[tex]$
\left[\begin{array}{lll}3 & 6 & 1 \\ 2 & 4 & 0 \\ 0 & 6 & 2\end{array}\right] \times \left[\begin{array}{l}2 \\ 0 \\ 1\end{array}\right]
$[/tex]
we need to perform the multiplication of the given 3x3 matrix with the 3x1 column vector. This operation is accomplished by taking the dot product of each row of the matrix with the column vector. Here's the step-by-step calculation:
1. First Row:
[tex]$
(3 \times 2) + (6 \times 0) + (1 \times 1)
$[/tex]
Simplifying, we get:
[tex]$
6 + 0 + 1 = 7
$[/tex]
2. Second Row:
[tex]$
(2 \times 2) + (4 \times 0) + (0 \times 1)
$[/tex]
Simplifying, we get:
[tex]$
4 + 0 + 0 = 4
$[/tex]
3. Third Row:
[tex]$
(0 \times 2) + (6 \times 0) + (2 \times 1)
$[/tex]
Simplifying, we get:
[tex]$
0 + 0 + 2 = 2
$[/tex]
Combining these results, the product matrix (or column vector) is:
[tex]$
\left[\begin{array}{c}7 \\ 4 \\ 2\end{array}\right]
$[/tex]
So, the product of the given matrices is:
[tex]$
\left[\begin{array}{c}7 \\ 4 \\ 2\end{array}\right]
$[/tex]
