Question

Find the product of the following two matrices
0 4 -1 -1
-3 0 0 -2

Answer 1

To find the product of the two matrices, we will multiply the first matrix by the second matrix.

Let \( A \) be the first matrix:

\[
A = \begin{pmatrix}
0 & 4 \\
-3 & 0
\end{pmatrix}
\]

And let \( B \) be the second matrix:

\[
B = \begin{pmatrix}
-1 & -1 \\
0 & -2
\end{pmatrix}
\]

To compute the product \( C = A \cdot B \), we'll use the formula for matrix multiplication, where the element \( c_{ij} \) in the resulting matrix \( C \) is calculated as:

\[
c_{ij} = \sum_{k} a_{ik} b_{kj}
\]

### Calculation of \( C \)

**Element \( c_{11} \)**:
\[
c_{11} = a_{11}b_{11} + a_{12}b_{21} = 0 \cdot (-1) + 4 \cdot 0 = 0
\]

**Element \( c_{12} \)**:
\[
c_{12} = a_{11}b_{12} + a_{12}b_{22} = 0 \cdot (-1) + 4 \cdot (-2) = 0 - 8 = -8
\]

**Element \( c_{21} \)**:
\[
c_{21} = a_{21}b_{11} + a_{22}b_{21} = (-3) \cdot (-1) + 0 \cdot 0 = 3 + 0 = 3
\]

**Element \( c_{22} \)**:
\[
c_{22} = a_{21}b_{12} + a_{22}b_{22} = (-3) \cdot (-1) + 0 \cdot (-2) = 3 + 0 = 3
\]

### Putting the elements together

Thus, the product matrix \( C \) is:

\[
C = \begin{pmatrix}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{pmatrix} = \begin{pmatrix}
0 & -8 \\
3 & 3
\end{pmatrix}
\]

Hence, the product of the two matrices is:

\[
\boxed{\begin{pmatrix}
0 & -8 \\
3 & 3
\end{pmatrix}}
\]