If `[(2,-1),(3,4)][(5,2),(7,4)]=[(3,x),(y,22)]`. Then the value of x + y is:

Options

A. 0
B. 43
C. 71
D. 25

Answer

To solve the given matrix equation \(\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix} = \begin{pmatrix} 3 & x \\ y & 22 \end{pmatrix}\), we will perform matrix multiplication on the left-hand side and then equate the resulting matrix to the right-hand side.

### Step 1: Matrix Multiplication
We will multiply the two matrices on the left-hand side:

\[ \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix} \]

To find the element in the first row and first column:

\[ (2 \cdot 5) + (-1 \cdot 7) = 10 - 7 = 3 \]

To find the element in the first row and second column:

\[ (2 \cdot 2) + (-1 \cdot 4) = 4 - 4 = 0 \]

To find the element in the second row and first column:

\[ (3 \cdot 5) + (4 \cdot 7) = 15 + 28 = 43 \]

To find the element in the second row and second column:

\[ (3 \cdot 2) + (4 \cdot 4) = 6 + 16 = 22 \]

So, the product of the two matrices is:

\[ \begin{pmatrix} 3 & 0 \\ 43 & 22 \end{pmatrix} \]

### Step 2: Equate the Matrices
Now we equate this result to the right-hand side of the equation:

\[ \begin{pmatrix} 3 & 0 \\ 43 & 22 \end{pmatrix} = \begin{pmatrix} 3 & x \\ y & 22 \end{pmatrix} \]

From this, we can compare corresponding elements:

1. From the first row, first column: \(3 = 3\) (no new information)
2. From the first row, second column: \(0 = x\)
3. From the second row, first column: \(43 = y\)
4. From the second row, second column: \(22 = 22\) (no new information)

### Step 3: Find Values of \(x\) and \(y\)
From the comparisons, we find:
- \(x = 0\)
- \(y = 43\)

### Step 4: Calculate \(x + y\)
Now, we can find \(x + y\):

\[ x + y = 0 + 43 = 43 \]

### Final Answer
Thus, the value of \(x + y\) is \(43\).

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