If a matrix X has (a+b) rows and (a+2) columns and a matrix Y has (b+1) rows and (a+3) columns If both XY and YX exist. Then what are the values of a,b respectively ?

Options

A. 3,2
B. 2,3
C. 2,4
D. 4,3

Answer

To solve the problem, we need to analyze the conditions under which the products of the matrices \( XY \) and \( YX \) exist.

### Step-by-step Solution:

1. **Identify the dimensions of matrices \( X \) and \( Y \)**:
- Matrix \( X \) has \( (a + b) \) rows and \( (a + 2) \) columns.
- Matrix \( Y \) has \( (b + 1) \) rows and \( (a + 3) \) columns.

2. **Condition for the product \( XY \) to exist**:
- The number of columns of \( X \) must equal the number of rows of \( Y \).
- Therefore, we have:
\[ a + 2 = b + 1 \]
- Rearranging this gives:
\[ a - b = -1 \quad \text{(Equation 1)} \]

3. **Condition for the product \( YX \) to exist**:
- The number of columns of \( Y \) must equal the number of rows of \( X \).
- Therefore, we have:
\[ a + 3 = a + b \]
- Simplifying this gives:
\[ 3 = b \quad \text{(Equation 2)} \]

4. **Substituting the value of \( b \) into Equation 1**:
- From Equation 2, we know \( b = 3 \).
- Substitute \( b \) into Equation 1:
\[ a - 3 = -1 \]
- Solving for \( a \) gives:
\[ a = 2 \]

5. **Final values**:
- Thus, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 3 \]

### Conclusion:
The values of \( a \) and \( b \) are \( 2 \) and \( 3 \) respectively.

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