To find the matrix $B$, we start with the equation provided:

$3 \begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix} - B = \begin{pmatrix} 5 & -9 \\ 7 & -2 \end{pmatrix}.$

Step 1: Multiply the scalar $3$ with the matrix $\begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}$

$3 \begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 9 & -3 \\ 3 & 0 \end{pmatrix}.$

Step 2: Substitute this result into the equation

$\begin{pmatrix} 9 & -3 \\ 3 & 0 \end{pmatrix} - B = \begin{pmatrix} 5 & -9 \\ 7 & -2 \end{pmatrix}.$

Step 3: Rearrange to isolate $B$

$B = \begin{pmatrix} 9 & -3 \\ 3 & 0 \end{pmatrix} - \begin{pmatrix} 5 & -9 \\ 7 & -2 \end{pmatrix}.$

Step 4: Perform the subtraction element-wise

$B = \begin{pmatrix} 9 - 5 & -3 - (-9) \\ 3 - 7 & 0 - (-2) \end{pmatrix}.$

$B = \begin{pmatrix} 4 & 6 \\ -4 & 2 \end{pmatrix}.$

Final Answer:

$B = \begin{pmatrix} 4 & 6 \\ -4 & 2 \end{pmatrix}.$

Would you like a detailed breakdown or verification of any step?

Here are related questions to explore:

1. How do you perform scalar multiplication on a matrix?
2. What is the process for adding or subtracting two matrices?
3. Can we solve similar problems with larger matrices?
4. How can matrices be applied in solving systems of linear equations?
5. What are common errors to watch for in matrix operations?

Tip: Always double