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 \( B \) and \( A \), we need to follow the rules of matrix multiplication. Let's denote the matrices as follows:

\[ A = \begin{pmatrix} 5 & -3 & 2 \\ -3 & 8 & 1 \end{pmatrix} \]
\[ B = \begin{pmatrix} 2 & 3 \\ 5 & 6 \\ 2 & -3 \end{pmatrix} \]

### Step 1: Verify Dimensions
Matrix \( A \) has dimensions \( 2 \times 3 \) (2 rows and 3 columns), and matrix \( B \) has dimensions \( 3 \times 2 \) (3 rows and 2 columns). The product \( BA \) is valid because the number of columns in \( B \) (2) matches the number of rows in \( A \) (3).

### Step 2: Calculate the Product
The resulting matrix \( BA \) will have dimensions \( 3 \times 3 \) (3 rows from \( B \) and 3 columns from \( A \)). We will calculate each element of the resulting matrix as follows:

1. **First Row of \( BA \)**:
- \( (2, 3) \cdot (5, -3, 2) = 2*5 + 3*(-3) = 10 - 9 = 1 \)
- \( (2, 3) \cdot (-3, 8, 1) = 2*(-3) + 3*8 = -6 + 24 = 18 \)

So, the first row is \( (1, 18) \).

2. **Second Row of \( BA \)**:
- \( (5, 6) \cdot (5, -3, 2) = 5*5 + 6*(-3) = 25 - 18 = 7 \)
- \( (5, 6) \cdot (-3, 8, 1) = 5*(-3) + 6*8 = -15 + 48 = 33 \)

So, the second row is \( (7, 33) \).

3. **Third Row of \( BA \)**:
- \( (2, -3) \cdot (5, -3, 2) = 2*5 + (-3)*(-3) = 10 + 9 = 19 \)
- \( (2, -3) \cdot (-3, 8, 1) = 2*(-3) + (-3)*8 = -6 - 24 = -30 \)

So, the third row is \( (19, -30) \).

### Step 3: Combine Results
Now, we can combine all the rows to form the resulting matrix \( BA \):

\[ BA = \begin{pmatrix} 1 & 18 \\ 7 & 33 \\ 19 & -30 \end{pmatrix} \]

### Final Result
Thus, the product \( BA \) is:

\[ BA = \begin{pmatrix} 1 & 18 \\ 7 & 33 \\ 19 & -30 \end{pmatrix} \]