If A = and and AB = I3, then x + y is equal to

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Special Type of Square Matrices

Question

If A = $[\begin{array}{ccc} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ and $B = [\begin{array}{ccc} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$ and AB = I₃, then x + y is equal to

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A = [\begin{array}{ccc} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

and

B = [\begin{array}{ccc} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

AB = [\begin{array}{ccc} 1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] [\begin{array}{ccc} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

\Rightarrow [\begin{array}{ccc} 1 + 0 + 0 & - 2 + 2 + 0 & y + 0 + x \\ 0 + 0 + 0 & 0 + 1 + 0 & 0 + 0 + 0 \\ 0 + 0 + 0 & 0 + 0 + 0 & 0 + 0 + 1 \end{array}] = I_{3}

where I is the 3 × 3 identity matrix.

\Rightarrow [\begin{array}{ccc} 1 & 0 & y + x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = \Rightarrow [\begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

Comparing, we get

y + x = 0

∴ x + y = 0