If A' = \left[\right. 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \left]\right.is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

Engineering

Mathematics

Multiplication of Matrices

Question

If A' = $[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{matrix}]$ is a matrix satisfying the equation AA^T = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to

(2, –1)

(–2, 1)

(2, 1)

(–2, –1)

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AA^T = 9 I

$= [\begin{matrix} 9 & 0 & a + 4 + 2 b \\ 0 & 9 & 0 \\ a + 4 + 2 b & 2 a + 2 - 2 b & a^{2} + 4 + b^{2} \end{matrix}]$

$[\begin{matrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{matrix}]$

$∴$ On solving, a + 2b = –4 2a – 2b = –2

We get (a = –2, b = –1)