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 matrices \(AB\) and \(BA\) can exist based on the dimensions of matrices \(A\) and \(B\).

### Step 1: Identify the dimensions of matrices A and B
Matrix \(A\) has \(x\) rows and \(x + 5\) columns, which gives it the dimensions \(x \times (x + 5)\).

Matrix \(B\) has \(y\) rows and \(11 - y\) columns, which gives it the dimensions \(y \times (11 - y)\).

### Step 2: Determine the conditions for matrix multiplication
For the product \(AB\) to exist, the number of columns in \(A\) must equal the number of rows in \(B\). Therefore:
\[ x + 5 = y \]

For the product \(BA\) to exist, the number of columns in \(B\) must equal the number of rows in \(A\). Therefore:
\[ 11 - y = x \]

### Step 3: Set up the equations
From the conditions above, we have two equations:
1. \(y = x + 5\)
2. \(x = 11 - y\)

### Step 4: Substitute the first equation into the second
Substituting \(y\) from the first equation into the second equation:
\[ x = 11 - (x + 5) \]
This simplifies to:
\[ x = 11 - x - 5 \]
\[ x + x = 11 - 5 \]
\[ 2x = 6 \]
\[ x = 3 \]

### Step 5: Find the value of y
Now, substitute \(x = 3\) back into the first equation 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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