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Given:
[tex]\[ A = \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ -2 \end{array}\right] \][/tex]
[tex]\[ B = \left[\begin{array}{llll} 6 & -7 & -1 & 8 \end{array}\right] \][/tex]

Find \( AB \) and \( BA \). If the answer is a \( 1 \times 1 \) matrix, just enter it as a number. Otherwise, enter it as a matrix.

[tex]\[ AB = \][/tex]
[tex]\[ BA = -14 \][/tex]

Sagot :

GinnyAnswer
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]
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