Matrix A has x rows and `x+5` columns. Matrix B has y rows and `11-y` columns. If both AB and BA exist, then the ordered pair (x, y) is

Options

A. (3, 3)
B. (8, 3)
C. (3, 8)
D. none of these

Answer

To solve the problem, we need to determine the ordered pair (x, y) such that both matrix products AB and BA exist. Let's break down the steps:

### Step 1: Understand the dimensions of matrices A and B
Matrix A has dimensions \( x \times (x + 5) \) and matrix B has dimensions \( y \times (11 - y) \).

### Step 2: Determine the conditions for the product AB to exist
For the product AB to exist, the number of columns in A must equal the number of rows in B. Therefore, we have:
\[ x + 5 = y \quad \text{(1)} \]

### Step 3: Determine the conditions for the product BA to exist
For the product BA to exist, the number of columns in B must equal the number of rows in A. Therefore, we have:
\[ 11 - y = x \quad \text{(2)} \]

### Step 4: Solve the system of equations
Now we have a system of two equations:
1. \( x + 5 = y \)
2. \( 11 - y = x \)

Substituting equation (1) into equation (2):
\[ 11 - (x + 5) = x \]
This simplifies to:
\[ 11 - x - 5 = x \]
\[ 6 = 2x \]
\[ x = 3 \]

### Step 5: Find the value of y
Now that we have \( x = 3 \), we can substitute this value back into equation (1) to find y:
\[ y = x + 5 = 3 + 5 = 8 \]

### Step 6: Write the ordered pair
Thus, the ordered pair \( (x, y) \) is:
\[ (x, y) = (3, 8) \]

### Final Answer:
The ordered pair \( (x, y) \) is \( (3, 8) \).

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