If A=pmatrix1&2&22&1&-2&2&bpmatrix is a matrix satisfying the equation AA^T=9I, where I is…

Question

If $A=\begin{pmatrix}1&2&2\2&1&-2\a&2&b\end{pmatrix}$ is a matrix satisfying the equation $AA^T=9I$, where $I$ is the $3	imes3$ identity matrix, then the ordered pair $(a,b)$ is equal to:

Accepted Answer

Computing $AA^T=9I$ and matching off-diagonal entries:

$(1,3)$-entry: $a+4+2b=0\Rightarrow a+2b=-4$.

$(2,3)$-entry: $2a+2-2b=0\Rightarrow a-b=-1$.

Solving: subtracting, $3b=-3\Rightarrow b=-1$, and $a=-2$.

$(a,b)=(-2,-1)$.

Solutions · 1