If A = [ * 20 c 1&2&2 2&1& - 2 &2&b ] is a matrix satisfying the … - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}1&2&2\\2&1&{ - 2}\\a&2&b\end{array}} \right]$ 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:

✓
$(-2,-1)$
B
$(2,-1)$
C
$(-2,1)$
D
$(2,1)$

✓

Answer

Correct option: A.

$(-2,-1)$

a
${{\rm{A}}{{\rm{A}}^ \top } = 9{\rm{I}}}$

$\left[ {\begin{array}{*{20}{c}}
1&2&2\\
2&1&{ - 2}\\
a&2&b
\end{array}} \right]\left[ {\begin{array}{*{20}{r}}
1&2&{\rm{a}}\\
2&1&2\\
2&{ - 2}&b
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
9&0&0\\
0&9&0\\
0&0&9
\end{array}} \right]$

${{\rm{a}} + 4 + 2{\rm{b}} = 0 \Rightarrow {\rm{a}} + 2{\rm{b}} = - 4}$ ......$(i)$

${2{\rm{a}} + 2 - 2{\rm{b}} = 0 \Rightarrow {\rm{a}} + 2{\rm{b}} = - 1}$ .......$(ii)$

${{\rm{ From (i) and (ii) }}}$

${3{\rm{b}} = - 3 \Rightarrow {\rm{b}} = - 1}$

${{\rm{a}} = - 2}$

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