If `A=[(5,-3,2),(-3,8,1)]` and `B=[(2,3),(5,6),(2,-3)]` , then find BA.

Options

A. `[(1,18,7),(7,33,16),(19,-30,1)]`
B. `[(-1,-19),(36,30)]`
C. `[(1,7,19),(18,33,1),(7,30,-10)]`
D. `[(36,-36),(19,-1)]`

Answer

To find the product of matrices \( BA \), we follow the matrix multiplication rules. Let's break it down step by step.

### Step 1: Identify the matrices
We have:
\[ A = \begin{pmatrix} 5 & -3 & 2 \\ -3 & 8 & 1 \end{pmatrix} \]
and
\[ B = \begin{pmatrix} 2 & 3 \\ 5 & 6 \\ 2 & -3 \end{pmatrix} \]

### Step 2: Check the dimensions for multiplication
Matrix \( B \) is a \( 3 \times 2 \) matrix, and matrix \( A \) is a \( 2 \times 3 \) matrix. The number of columns in \( B \) (2) matches the number of rows in \( A \) (2), so we can multiply them. The resulting matrix \( BA \) will have the dimensions of \( 3 \times 3 \).

### Step 3: Multiply the matrices
To find \( BA \), we will compute each element of the resulting \( 3 \times 3 \) matrix. The element in the \( i^{th} \) row and \( j^{th} \) column of \( BA \) is calculated by taking the dot product of the \( i^{th} \) row of \( B \) and the \( j^{th} \) column of \( A \).

1. **First row, first column**:
\[ (2)(5) + (3)(-3) = 10 - 9 = 1 \]

2. **First row, second column**:
\[ (2)(-3) + (3)(8) = -6 + 24 = 18 \]

3. **First row, third column**:
\[ (2)(2) + (3)(1) = 4 + 3 = 7 \]

4. **Second row, first column**:
\[ (5)(5) + (6)(-3) = 25 - 18 = 7 \]

5. **Second row, second column**:
\[ (5)(-3) + (6)(8) = -15 + 48 = 33 \]

6. **Second row, third column**:
\[ (5)(2) + (6)(1) = 10 + 6 = 16 \]

7. **Third row, first column**:
\[ (2)(5) + (-3)(-3) = 10 + 9 = 19 \]

8. **Third row, second column**:
\[ (2)(-3) + (-3)(8) = -6 - 24 = -30 \]

9. **Third row, third column**:
\[ (2)(2) + (-3)(1) = 4 - 3 = 1 \]

### Step 4: Compile the resulting matrix
Putting all the computed values together, we have:
\[ BA = \begin{pmatrix} 1 & 18 & 7 \\ 7 & 33 & 16 \\ 19 & -30 & 1 \end{pmatrix} \]

### Final Result
Thus, the product \( BA \) is:
\[ BA = \begin{pmatrix} 1 & 18 & 7 \\ 7 & 33 & 16 \\ 19 & -30 & 1 \end{pmatrix} \]