If A = `[{:(1),(-4),(3):}]` and B = [-1, 2, 1], then (AB)' is equal to

Options

A. `[{:(-1, 4, -3),(2, - 8, 6),(1, - 4, 3):}] `
B. `[{:(-1, 2, 1),(4, - 8, -4),(-3, 6, 3):}] `
C. `[{:(1, 4, -3),(2, - 8, 6),(1, 4, 3):}] `
D. `[{:(-1, 4, -3),(2, 8, 6),(1, - 4, 3):}] `

Answer

To solve the problem, we need to find the product of matrices A and B, and then take the transpose of that product. Let's break down the steps:

### Step 1: Identify the matrices A and B
Given:
- Matrix \( A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix} \) (a column matrix of order 3x1)
- Matrix \( B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix} \) (a row matrix of order 1x3)

### Step 2: Check the dimensions for multiplication
Matrix \( A \) is of size 3x1 and matrix \( B \) is of size 1x3. The multiplication \( AB \) is possible because the number of columns in \( A \) (1) is equal to the number of rows in \( B \) (1).

### Step 3: Perform the matrix multiplication \( AB \)
To find \( AB \), we multiply the column matrix \( A \) by the row matrix \( B \):

\[ AB = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix} \begin{bmatrix} -1 & 2 & 1 \end{bmatrix} \]

Calculating the product:

\[ AB = \begin{bmatrix} 1 \cdot -1 & 1 \cdot 2 & 1 \cdot 1 \\ -4 \cdot -1 & -4 \cdot 2 & -4 \cdot 1 \\ 3 \cdot -1 & 3 \cdot 2 & 3 \cdot 1 \end{bmatrix} \]

This results in:

\[ AB = \begin{bmatrix} -1 & 2 & 1 \\ 4 & -8 & -4 \\ -3 & 6 & 3 \end{bmatrix} \]

### Step 4: Take the transpose of the product \( AB \)
The transpose of a matrix is obtained by swapping its rows and columns. Therefore, the transpose \( (AB)' \) is:

\[ (AB)' = \begin{bmatrix} -1 & 4 & -3 \\ 2 & -8 & 6 \\ 1 & -4 & 3 \end{bmatrix} \]

### Final Answer
Thus, the result of \( (AB)' \) is:

\[ (AB)' = \begin{bmatrix} -1 & 4 & -3 \\ 2 & -8 & 6 \\ 1 & -4 & 3 \end{bmatrix} \]

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