Given the matrices \(A\) and \(B\):
[tex]\[A = \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ -2 \end{array}\right]\][/tex]
[tex]\[B = \left[\begin{array}{cccc} 6 & -7 & -1 & 8 \end{array}\right]\][/tex]
We need to find the products \(AB\) and \(BA\):
### Step-by-Step Solution
1. Matrix Multiplication \(AB\):
- \(A\) is a \(4 \times 1\) matrix.
- \(B\) is a \(1 \times 4\) matrix.
The product \(AB\) will result in a \(4 \times 4\) matrix because the outer dimensions determine the dimensions of the resulting matrix.
The elements of the resulting matrix \(AB\) are computed as follows:
[tex]\[
AB = A \cdot B = \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ -2 \end{array}\right] \cdot \left[\begin{array}{cccc} 6 & -7 & -1 & 8 \end{array}\right] = \left[\begin{array}{cccc}
1 \cdot 6 & 1 \cdot (-7) & 1 \cdot (-1) & 1 \cdot 8 \\
0 \cdot 6 & 0 \cdot (-7) & 0 \cdot (-1) & 0 \cdot 8 \\
4 \cdot 6 & 4 \cdot (-7) & 4 \cdot (-1) & 4 \cdot 8 \\
-2 \cdot 6 & -2 \cdot (-7) & -2 \cdot (-1) & -2 \cdot 8 \\
\end{array}\right]
\][/tex]
Simplify each element:
[tex]\[
AB = \left[\begin{array}{cccc}
6 & -7 & -1 & 8 \\
0 & 0 & 0 & 0 \\
24 & -28 & -4 & 32 \\
-12 & 14 & 2 & -16
\end{array}\right]
\][/tex]
2. Matrix Multiplication \(BA\):
- \(B\) is a \(1 \times 4\) matrix.
- \(A\) is a \(4 \times 1\) matrix.
The product \(BA\) will result in a \(1 \times 1\) matrix, which is essentially a single number.
The element of the resulting matrix \(BA\) is computed as follows:
[tex]\[
BA = B \cdot A = \left[\begin{array}{cccc} 6 & -7 & -1 & 8 \end{array}\right] \cdot \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ -2 \end{array}\right] = 6 \cdot 1 + (-7 \cdot 0) + (-1 \cdot 4) + (8 \cdot -2)
\][/tex]
Simplify the expression:
[tex]\[
BA = 6 + 0 - 4 - 16 = -14
\][/tex]
Therefore, the final solutions are:
[tex]\[ A B = \left[\begin{array}{cccc}
6 & -7 & -1 & 8 \\
0 & 0 & 0 & 0 \\
24 & -28 & -4 & 32 \\
-12 & 14 & 2 & -16
\end{array}\right] \][/tex]
[tex]\[ B A = -14 \][/tex]
