Question

If

\[ X = \begin{bmatrix} 3 & 4 \\ 2 & -1 \end{bmatrix}, \quad 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

then find the matrix \( Y \).

Hint:

When solving matrix equations, isolate the variable matrix and perform matrix operations like addition or subtraction element-wise.

Answer 1

Step 1: Isolate \( Y \).

We are given the equation:

\[ 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

Rearranging to solve for \( Y \):

\[ Y = 2X - \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

Step 2: Calculate \( 2X \).

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

Step 3: Subtract the matrices.

\[ Y = \begin{bmatrix} 6 & 8 \\ 4 & -2 \end{bmatrix} - \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]

\[ Y = \begin{bmatrix} 6 - 5 & 8 - 10 \\ 4 - 3 & -2 - (-5) \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 1 & 3 \end{bmatrix} \]

Step 4: Conclusion.

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