If `A^(-1)=[(0,1,-2),(-2,9,-23),(-1,5,-13)],B^(T)=[11-5-3]` and `X=A^(-1)B`, then X=

Options

A. [1, 2,3]
B. `[(3),(2),(1)]`
C. [3,2,1]
D. `[(1),(2),(3)]`

Answer

To solve the problem, we need to find the matrix \( X \) using the formula \( X = A^{-1}B \). We are given \( A^{-1} \) and \( B^T \). Let's break it down step by step.

### Step 1: Identify the matrices
We have:
\[ A^{-1} = \begin{pmatrix} 0 & 1 & -2 \\ -2 & 9 & -23 \\ -1 & 5 & -13 \end{pmatrix} \]
and
\[ B^T = \begin{pmatrix} 11 \\ -5 \\ -3 \end{pmatrix} \]
To find \( B \), we need to transpose \( B^T \):
\[ B = \begin{pmatrix} 11 & -5 & -3 \end{pmatrix} \]

### Step 2: Set up the multiplication
Now we need to calculate \( X = A^{-1}B \). The dimensions of \( A^{-1} \) are \( 3 \times 3 \) and the dimensions of \( B \) are \( 3 \times 1 \). The resulting matrix \( X \) will be of dimensions \( 3 \times 1 \).

### Step 3: Perform the matrix multiplication
We will multiply each row of \( A^{-1} \) by the column of \( B \):

1. **First row calculation**:
\[ X_1 = (0 \times 11) + (1 \times -5) + (-2 \times -3) = 0 - 5 + 6 = 1 \]

2. **Second row calculation**:
\[ X_2 = (-2 \times 11) + (9 \times -5) + (-23 \times -3) = -22 - 45 + 69 = 2 \]

3. **Third row calculation**:
\[ X_3 = (-1 \times 11) + (5 \times -5) + (-13 \times -3) = -11 - 25 + 39 = 3 \]

### Step 4: Compile the results
Thus, we have:
\[ X = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]

### Final Answer
The final result is:
\[ X = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]