If `[(x+y,a+b),(a-b,x-y)]=[(5-1),(3-5)]` the the values of x, y, a, b are respectively.

Options

A. 0,-5,1,2
B. 0,5,1,-2
C. 0,5,-1,2
D. 0,-5,1,-2

Answer

To solve the problem step-by-step, we start with the given matrix equation:

\[ \begin{pmatrix} x+y & a+b \\ a-b & x-y \end{pmatrix} = \begin{pmatrix} 5 & -1 \\ 3 & -5 \end{pmatrix} \]

### Step 1: Set up the equations from the matrix

From the matrix equality, we can equate the corresponding elements:

1. \( x + y = 5 \) (Equation 1)
2. \( a + b = -1 \) (Equation 2)
3. \( a - b = 3 \) (Equation 3)
4. \( x - y = -5 \) (Equation 4)

### Step 2: Solve for \(x\) and \(y\)

From Equation 1 and Equation 4, we can solve for \(x\) and \(y\).

- From Equation 1:
\[ x + y = 5 \quad \text{(1)} \]

- From Equation 4:
\[ x - y = -5 \quad \text{(4)} \]

Now, we can add these two equations to eliminate \(y\):

\[ (x + y) + (x - y) = 5 + (-5) \]
\[ 2x = 0 \]
\[ x = 0 \]

Now, substitute \(x = 0\) back into Equation 1 to find \(y\):

\[ 0 + y = 5 \]
\[ y = 5 \]

### Step 3: Solve for \(a\) and \(b\)

Next, we will solve for \(a\) and \(b\) using Equation 2 and Equation 3.

- From Equation 2:
\[ a + b = -1 \quad \text{(2)} \]

- From Equation 3:
\[ a - b = 3 \quad \text{(3)} \]

Now, we can add these two equations to eliminate \(b\):

\[ (a + b) + (a - b) = -1 + 3 \]
\[ 2a = 2 \]
\[ a = 1 \]

Now, substitute \(a = 1\) back into Equation 2 to find \(b\):

\[ 1 + b = -1 \]
\[ b = -1 - 1 = -2 \]

### Final Values

We have found the values:
- \(x = 0\)
- \(y = 5\)
- \(a = 1\)
- \(b = -2\)

Thus, the values of \(x\), \(y\), \(a\), and \(b\) are respectively \(0\), \(5\), \(1\), and \(-2\).

### Final Answer

The final answer is:
\[ \text{Values of } x, y, a, b \text{ are } 0, 5, 1, -2 \]

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