Question

Matrix \( A \) has \( x \) rows and \( x + 5 \) columns. Matrix \( B \) has \( y \) rows and \( 11 - y \) columns. Both \( AB \) and \( BA \) exist, then

• A. \( x = 3 \) and \( y = 8 \)
• B. \( x = 3 \) and \( y = 4 \)
• C. \( x = 4 \) and \( y = 8 \)
• D. \( x = 5 \) and \( y = 9 \)

Hint:

Matrix multiplication \( AB \) is defined only if columns of \( A \) = rows of \( B \); for \( BA \), it's the reverse.

Answer 1

The Correct Option is A

Matrix \( A \): \( x \times (x + 5) \) Matrix \( B \): \( y \times (11 - y) \) For \( AB \) to exist: Columns of \( A \) = Rows of \( B \) \[ x + 5 = y \tag{1} \] For \( BA \) to exist: Columns of \( B \) = Rows of \( A \) \[ 11 - y = x \tag{2} \] Solving equations (1) and (2): From (1): \( y = x + 5 \) Substitute in (2): \[ 11 - (x + 5) = x \Rightarrow 6 = x \Rightarrow x = 3, y = 8 \]