Question

Find the product of the 2 matrices
3 2 1 4
5 -4 -3 2
-1 -3

Answer 1

To find the product of the two matrices, we first need to define the matrices clearly. Let's denote the first matrix as \( A \) and the second matrix as \( B \):

\[
A = \begin{pmatrix}
3 & 2 \\
5 & -4 \\
-1 & -3
\end{pmatrix}
\]

\[
B = \begin{pmatrix}
1 & 4 \\
-3 & 2
\end{pmatrix}
\]

Matrix \( A \) has a size of \( 3 \times 2 \) (3 rows and 2 columns), and matrix \( B \) has a size of \( 2 \times 2 \) (2 rows and 2 columns). The product \( AB \) will result in a \( 3 \times 2 \) matrix, since the number of rows comes from the first matrix \( A \) and the number of columns comes from the second matrix \( B \).

To compute the matrix product \( AB \), we use the formula:
\[
(AB)_{i,j} = \sum_{k} A_{i,k} B_{k,j}
\]
where \( i \) is the row index of matrix \( A \), \( j \) is the column index of matrix \( B \), and \( k \) iterates over the columns of \( A \) (which are the same as the rows of \( B \)).

Now let's compute each element of the resulting matrix \( AB \):

1. **Row 1, Column 1**:
\[
(AB)_{1,1} = 3 \cdot 1 + 2 \cdot (-3) = 3 - 6 = -3
\]

2. **Row 1, Column 2**:
\[
(AB)_{1,2} = 3 \cdot 4 + 2 \cdot 2 = 12 + 4 = 16
\]

3. **Row 2, Column 1**:
\[
(AB)_{2,1} = 5 \cdot 1 + (-4) \cdot (-3) = 5 + 12 = 17
\]

4. **Row 2, Column 2**:
\[
(AB)_{2,2} = 5 \cdot 4 + (-4) \cdot 2 = 20 - 8 = 12
\]

5. **Row 3, Column 1**:
\[
(AB)_{3,1} = -1 \cdot 1 + (-3) \cdot (-3) = -1 + 9 = 8
\]

6. **Row 3, Column 2**:
\[
(AB)_{3,2} = -1 \cdot 4 + (-3) \cdot 2 = -4 - 6 = -10
\]

Putting it all together, we have the product matrix:

\[
AB = \begin{pmatrix}
-3 & 16 \\
17 & 12 \\
8 & -10
\end{pmatrix}
\]

Thus, the product of the two matrices \( A \) and \( B \) is:

\[
\begin{pmatrix}
-3 & 16 \\
17 & 12 \\
8 & -10
\end{pmatrix}
\]