The problem involves determining if the product $AB$ of the two given matrices $A$ and $B$ can be computed, and if so, calculating the resulting matrix.

Step 1: Dimensions of Matrices

• Matrix $A$ is $1 \times 3$ (1 row, 3 columns).
• Matrix $B$ is $3 \times 3$ (3 rows, 3 columns).

Step 2: Compatibility for Matrix Multiplication

Matrix multiplication $AB$ is possible if the number of columns in $A$ matches the number of rows in $B$.

• Here, $A$ has 3 columns, and $B$ has 3 rows.
• Therefore, matrix multiplication is possible.

Step 3: Dimensions of the Resulting Matrix

The resulting matrix $AB$ will have dimensions $1 \times 3$ (rows of $A$ by columns of $B$).

Step 4: Perform Matrix Multiplication

We compute $AB$ as follows:

$A = \begin{bmatrix} 5 & 2 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 9 & 3 & 3 \\ -2 & -7 & 0 \\ 4 & 0 & 7 \end{bmatrix}$