If `A^T=[(3,4),(-1,2),(0,1)]` and B=`[(-1,2,1),(1,2,3)]`, then find `A^T-B^T`.

Options

A. `[(4,3),(-3,0),(-1,-2)]`
B. `[(4,3),(3,0),(1,2)]`
C. `[(4,0),(-1,-3),(3,-2)]`
D. `[(1,-3),(2,4),(3,5)]`

Answer

To solve the problem of finding \( A^T - B^T \), we will follow these steps:

### Step 1: Identify the matrices \( A^T \) and \( B \)
Given:
\[ A^T = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \]
\[ B = \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} \]

### Step 2: Find the transpose of matrix \( B \)
To find \( B^T \), we will convert rows of \( B \) into columns:
\[ B^T = \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} \]

### Step 3: Prepare to subtract \( B^T \) from \( A^T \)
Now we have:
\[ A^T = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \]
\[ B^T = \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} \]

### Step 4: Perform the subtraction \( A^T - B^T \)
We will subtract corresponding elements of \( B^T \) from \( A^T \):
\[ A^T - B^T = \begin{pmatrix} 3 - (-1) & 4 - 1 \\ -1 - 2 & 2 - 2 \\ 0 - 1 & 1 - 3 \end{pmatrix} \]

Calculating each element:
- First row:
- \( 3 - (-1) = 3 + 1 = 4 \)
- \( 4 - 1 = 3 \)

- Second row:
- \( -1 - 2 = -3 \)
- \( 2 - 2 = 0 \)

- Third row:
- \( 0 - 1 = -1 \)
- \( 1 - 3 = -2 \)

Putting it all together, we get:
\[ A^T - B^T = \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]

### Final Result
Thus, the result of \( A^T - B^T \) is:
\[ \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]

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