Question

If a=12a21222b is a matrix satisfying the equation aat=9i, where i is 33 identity matrix, then the ordered pair (a,b) is equal to

Answer 1

Answer:

Values of a are 2, -2

values of b are -3, 1, -1

Step-by-step explanation:

$A=\left[\begin{array}{ccc}1&2&a\\2&1&2\\2&2&b\end{array}\right] \\\\\\A^T=\left[\begin{array}{ccc}1&2&2\\2&1&2\\a&2&b\end{array}\right]$

$Given:\\\\AA^T=9I\\\\\left[\begin{array}{ccc}1&2&a\\2&1&2\\2&2&b\end{array}\right] \left[\begin{array}{ccc}1&2&2\\2&1&2\\a&2&b\end{array}\right]=9\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]\\\\\\\left[\begin{array}{ccc}5+a^2&24+2a&6+ab\\4+2a&9&6+2b\\6+ab&6+2b&8+b^2\end{array}\right]=\left[\begin{array}{ccc}9&0&0\\0&9&0\\0&0&9\end{array}\right]$

Equating the corresponding elements on both sides, we get

4+2a=0 ⇒ 2a=-4 ⇒ a=-2

6+ab=0 ⇒6+(-2b)=0 ⇒ 2b=6 ⇒b=3

$5+a^2=9 \\\\a^2=4\\\\a=2, -2$

6+2b=0 ⇒ 2b = -6 ⇒ b=-3

$8+b^2=9\\\\b^2=1\\\\b=1,-1$