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EEJ MAIN Mathematics QUESTION #6941
Question 1

Matrix $A = \begin{pmatrix}1&2&2\\2&1&-2\\a&2&b\end{pmatrix}$ satisfies $AA^T = 9I$, where $I$ is the $3\times3$ identity matrix. Find the ordered pair $(a, b)$.

  • $(2,-1)$
  • $(-2,1)$
  • $(2,1)$
  • $(-2,-1)$✔️
Correct Answer Explanation

$AA^T = 9I$ means each row has magnitude 3 and rows are mutually orthogonal.

Row 1: $1^2+2^2+2^2=9$ ✓

Row 2: $4+1+4=9$ ✓

Row 3 magnitude: $a^2+4+b^2=9 \Rightarrow a^2+b^2=5$ ...(i)

Row 1 · Row 3: $a+4+2b=0 \Rightarrow a+2b=-4$ ...(ii)

Row 2 · Row 3: $2a+2-2b=0 \Rightarrow a-b=-1$ ...(iii)

From (ii) and (iii): subtracting gives $3b=-3 \Rightarrow b=-1$, then $a=-2$. Check (i): $4+1=5$ ✓

$(a,b) = \mathbf{(-2,-1)}$