Let's solve the given matrix equation step by step. The equation provided is:

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

Step 1: Simplify the equation

Multiply through by $-3$ to eliminate the fractions:

$X - A = 3B$

Rearranging for $X$:

$X = A + 3B$

Step 2: Substitute the given matrices

The matrices are: $A = \begin{bmatrix} 9 & -6 \\ 3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} -9 & -12 \\ 12 & 10 \end{bmatrix}.$

Substituting, we have: $X = A + 3B.$

Step 3: Calculate $3B$

Multiply $B$ by 3: $3B = 3 \cdot \begin{bmatrix} -9 & -12 \\ 12 & 10 \end{bmatrix} = \begin{bmatrix} -27 & -36 \\ 36 & 30 \end{bmatrix}.$

Step 4: Add $A$ and $3B$

Now add the matrices $A$ and $3B$: $X = \begin{bmatrix} 9 & -6 \\ 3 & 6 \end{bmatrix} + \begin{bmatrix} -27 & -36 \\ 36 & 30 \end{bmatrix}.$

Perform the addition element-wise: $X = \begin{bmatrix} 9 + (-27) & -6 + (-36) \\ 3 + 36 & 6 + 30 \end{bmatrix} = \begin{bmatrix} -18 & -42 \\ 39 & 36 \end{bmatrix}.$

Final Answer:

$X = \begin{bmatrix} -18 & -42 \\ 39 & 36 \end{bmatrix}.$

Would you like to review any specific step? Here are some related questions for practice:

1. How do you compute the scalar multiplication of a matrix?
2. What are the properties of matrix addition and subtraction?
3. How can we solve matrix equations with more than one variable?
4. What is the determinant of the resulting matrix $X$?
5. Can this method be extended to non-square matrices?

Tip: Always ensure the dimensions of matrices align correctly during addition and multiplication!