If \(\mathrm{A} = \begin{bmatrix} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\), \(\mathrm{B} = \begin{bmatrix} 1 & -2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) and AB = I3, where I3 is the unit matrix of order 3 × 3, then \(x + y\) is equal to:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
Each entry of AB is a row of A times a column of B. Only the first row of A contains x, so only the first row of AB needs working:
\((1, 2, x)\cdot(1, 0, 0) = 1\)
\((1, 2, x)\cdot(-2, 1, 0) = -2 + 2 = 0\)
\((1, 2, x)\cdot(y, 0, 1) = y + x\)
The second and third rows of AB are (0, 1, 0) and (0, 0, 1), so
\( \displaystyle \mathrm{AB} = \begin{bmatrix} 1 & 0 & x + y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
For AB to be the unit matrix I3, the entry in row 1, column 3 must be 0:
\(x + y = 0\)