Question

If \( X + Y = \begin{pmatrix} 5 & 3 \\ 0 & 7 \end{pmatrix} \) and \( X - Y = \begin{pmatrix} 7 & 1 2 & 3 \end{pmatrix} \). Then X and Y are

• A. \( X = \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} \text{ and } Y = \begin{pmatrix} -1 & 1 \\ -1 & 2 \end{pmatrix} \)
• B. \( X = \begin{pmatrix} 0 & 5 \\ 1 & 4 \end{pmatrix} \text{ and } Y = \begin{pmatrix} -1 & -1 \\ -1 & 2 \end{pmatrix} \)
• C. \( X = \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} \text{ and } Y = \begin{pmatrix} -1 & -1 \\ -1 & 2 \end{pmatrix} \)
• D. \( X = \begin{pmatrix} 1 & 4 \\ 5 & 0 \end{pmatrix} \text{ and } Y = \begin{pmatrix} 0 & 2 \\ 1 & 1 \end{pmatrix} \)

Hint:

Treat matrix equations like scalar equations for addition/subtraction. Add or subtract equations to isolate one variable, then back-substitute. Always perform element-wise addition/subtraction carefully. Double-check your result against options to catch any simple arithmetic errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question provides two matrix equations involving two unknown matrices X and Y. We need to solve for these matrices.

Step 2: Key Formula or Approach:
This is analogous to solving a system of linear equations for scalar variables. We can use addition and subtraction of matrix equations.
Given:
1. \( X + Y = A = \begin{pmatrix} 5 & 3 \\ 0 & 7 \end{pmatrix} \)
2. \( X - Y = B = \begin{pmatrix} 7 & 1 \\ 2 & 3 \end{pmatrix} \)

Step 3: Detailed Explanation:
Solve for X:
Add Equation 1 and Equation 2:
\( (X + Y) + (X - Y) = A + B \)
\( 2X = \begin{pmatrix} 5 & 3 \\ 0 & 7 \end{pmatrix} + \begin{pmatrix} 7 & 1 \\ 2 & 3 \end{pmatrix} \)
Perform matrix addition (add corresponding elements):
\( 2X = \begin{pmatrix} 5+7 & 3+1 \\ 0+2 & 7+3 \end{pmatrix} = \begin{pmatrix} 12 & 4 \\ 2 & 10 \end{pmatrix} \)
Now, divide by 2 (multiply by 1/2):
\( X = \frac{1}{2} \begin{pmatrix} 12 & 4 \\ 2 & 10 \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} \)
Solve for Y:
Substitute the value of X back into Equation 1:
\( X + Y = A \)
\( \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} + Y = \begin{pmatrix} 5 & 3 \\ 0 & 7 \end{pmatrix} \)
Subtract X from both sides:
\( Y = \begin{pmatrix} 5 & 3 \\ 0 & 7 \end{pmatrix} - \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} \)
Perform matrix subtraction:
\( Y = \begin{pmatrix} 5-6 & 3-2 \\ 0-1 & 7-5 \end{pmatrix} = \begin{pmatrix} -1 & 1 \\ -1 & 2 \end{pmatrix} \)

Step 4: Final Answer:
\( X = \begin{pmatrix} 6 & 2 \\ 1 & 5 \end{pmatrix} \text{ and } Y = \begin{pmatrix} -1 & 1 \\ -1 & 2 \end{pmatrix} \). This corresponds to option (a).