Question

If \( X = \begin{bmatrix} 3 & 1 \\ 2 & -1 \end{bmatrix} \) and \( 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \), then find the matrix \( Y \).

Hint:

Subtract corresponding elements carefully. Pay close attention to double negatives, such as \( -2 - (-5) = -2 + 5 = 3 \).

Answer 1

Step 1: Understanding the Concept:
Matrix algebra allows us to perform scalar multiplication and subtraction. If \( 2X - Y = A \), then \( Y = 2X - A \).
Step 2: Detailed Explanation:
Given \( X = \begin{bmatrix} 3 & 1 \\ 2 & -1 \end{bmatrix} \).
First, find \( 2X \):
\[ 2X = 2 \begin{bmatrix} 3 & 1 \\ 2 & -1 \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 4 & -2 \end{bmatrix} \]
Now, given \( 2X - Y = \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \).
Isolate \( Y \):
\[ Y = 2X - \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]
\[ Y = \begin{bmatrix} 6 & 2 \\ 4 & -2 \end{bmatrix} - \begin{bmatrix} 5 & 10 \\ 3 & -5 \end{bmatrix} \]
\[ Y = \begin{bmatrix} 6-5 & 2-10 \\ 4-3 & -2-(-5) \end{bmatrix} = \begin{bmatrix} 1 & -8 \\ 1 & 3 \end{bmatrix} \]
Step 3: Final Answer:
The matrix \( Y \) is \( \begin{bmatrix} 1 & -8 \\ 1 & 3 \end{bmatrix} \).