We are solving for $X$ in the matrix equation:

$3X - \frac{1}{3}A = B$

Step 1: Rearrange the equation

Isolate $3X$: $3X = B + \frac{1}{3}A$

Divide through by 3 to solve for $X$: $X = \frac{1}{3}(B + \frac{1}{3}A)$

Step 2: Substitute the given matrices

The matrices $A$ and $B$ are: $A = \begin{bmatrix} -3 & -6 \\ -3 & -9 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 4 \\ 9 & 9 \end{bmatrix}$

First, compute $\frac{1}{3}A$: $\frac{1}{3}A = \frac{1}{3} \begin{bmatrix} -3 & -6 \\ -3 & -9 \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ -1 & -3 \end{bmatrix}$

Now, add $B$ and $\frac{1}{3}A$: $B + \frac{1}{3}A = \begin{bmatrix} 7 & 4 \\ 9 & 9 \end{bmatrix} + \begin{bmatrix} -1 & -2 \\ -1 & -3 \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 8 & 6 \end{bmatrix}$

Finally, divide by 3 to find $X$: $X = \frac{1}{3} \begin{bmatrix} 6 & 2 \\ 8 & 6 \end{bmatrix} = \begin{bmatrix} 2 & \frac{2}{3} \\ \frac{8}{3} & 2 \end{bmatrix}$

Final Answer:

$X = \begin{bmatrix} 2 & \frac{2}{3} \\ \frac{8}{3} & 2 \end{bmatrix}$

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Would you like a breakdown of each calculation or more details on the steps? Here are 5 related questions to explore further:

1. How can we verify the solution of a matrix equation?
2. What are the rules for scalar multiplication and addition in matrices?
3. Can this approach be generalized for non-square matrices $A$ and $B$?
4. How does matrix division differ from scalar division in solving equations?
5. How would the solution change if $A$ or $B$ were singular matrices?

Tip: Always double-check matrix dimensions to ensure that addition, subtraction, and multiplication operations are valid.