9514 1404 393
Answer:
(a, b) = (-2, -1)
Step-by-step explanation:
The transpose of the given matrix is ...
[tex]A^T=\left[\begin{array}{ccc}1&2&a\\2&1&2\\2&-2&b\end{array}\right][/tex]
Then the [3,1] term of the product is ...
[tex](A\cdot A^T)_{31}=\left[\begin{array}{ccc}a&2&b\end{array}\right]\cdot\left[\begin{array}{ccc}1&2&2\end{array}\right]=a+2b+4[/tex]
and the [3,2] term is ...
[tex](A\cdot A^T)_{32}=\left[\begin{array}{ccc}a&2&b\end{array}\right]\cdot\left[\begin{array}{ccc}2&1&-2\end{array}\right]=2a-2b+2[/tex]
Both of these terms in the product matrix are 0. We can solve the system of equations by adding these two terms:
(a +2b +4) +(2a -2b +2) = (0) +(0)
3a +6 = 0
a = -2
Substituting for 'a' in term [3,1] gives ...
-2 +2b +4 = 0
b = -1
The ordered pair (a, b) is (-2, -1).
