If `X'=[(3,4),(-1,2),(0,1)]` and `Y=[(-1,2,1),(1,2,3)]`, then find `X'-Y'`

Answer

To solve the problem of finding \( X' - Y' \), we first need to determine the transposes of matrices \( X \) and \( Y \).

Given:
\[ X' = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \]
\[ Y = \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} \]

### Step 1: Find the transpose of matrix \( Y \)

The transpose of a matrix is obtained by swapping its rows and columns. Therefore, the transpose \( Y' \) of matrix \( Y \) will be:
\[ Y' = \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} \]

### Step 2: Perform the subtraction \( X' - Y' \)

Now we will subtract \( Y' \) from \( X' \):
\[ X' - Y' = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} - \begin{pmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{pmatrix} \]

To perform the subtraction, we subtract corresponding elements:
\[ = \begin{pmatrix} 3 - (-1) & 4 - 1 \\ -1 - 2 & 2 - 2 \\ 0 - 1 & 1 - 3 \end{pmatrix} \]

Calculating each element:
1. First row: \( 3 - (-1) = 3 + 1 = 4 \) and \( 4 - 1 = 3 \)
2. Second row: \( -1 - 2 = -3 \) and \( 2 - 2 = 0 \)
3. Third row: \( 0 - 1 = -1 \) and \( 1 - 3 = -2 \)

Thus, we have:
\[ X' - Y' = \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]

### Final Answer
\[ X' - Y' = \begin{pmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{pmatrix} \]