Question

If \( 2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \) and \( Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \), then \( X \) will be

• A. \( \begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix} \)
• B. \( \begin{bmatrix} -1 & -1 \\ -1 & -2 \end{bmatrix} \)
• C. \( \begin{bmatrix} -2 & -1 \\ -1 & -1 \end{bmatrix} \)
• D. \( \begin{bmatrix} 1 & -1 \\ -1 & -1 \end{bmatrix} \)

Hint:

When solving matrix equations, always perform the addition or subtraction first before dealing with scalar multiplication or division. This helps prevent calculation errors. Double-check your arithmetic, especially with negative numbers.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a matrix algebra problem. We need to solve for the unknown matrix \(X\) by treating the equation like a standard algebraic equation, using matrix subtraction and scalar multiplication.
Step 2: Key Formula or Approach:
The given equation is \( 2X + Y = A \), where \( A = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} \).
To find \(X\), isolate \(2X\): \[ 2X = A - Y \] Then solve for \(X\) by multiplying by \(\frac{1}{2}\): \[ X = \frac{1}{2}(A - Y) \] Step 3: Detailed Explanation:
Substitute the given matrices: \[ 2X = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix} - \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} = \begin{bmatrix} 1-3 & 0-2 \\ -3-1 & 2-4 \end{bmatrix} = \begin{bmatrix} -2 & -2 \\ -4 & -2 \end{bmatrix} \] Multiply by \(\frac{1}{2}\) to get \(X\): \[ X = \frac{1}{2} \begin{bmatrix} -2 & -2 \\ -4 & -2 \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix} \] Step 4: Final Answer:
The matrix \(X\) is \(\boxed{\begin{bmatrix} -1 & -1 \\ -2 & -1 \end{bmatrix}}\). This corresponds to option (A).