Matrices and Determinants 3 Question 13
15. If the adjoint of a $3 \times 3$ matrix $P$ is $2 \quad 1 \quad 7$, then the
$\begin{bmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{bmatrix}$ possible value(s) of the determinant of $P$ is/are
(a) -2
(b) -1
(c) 1
(d) 2
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Answer:
Correct Answer: 15. (a,d)
Solution:
- PLAN: If $\left|A_{n \times n}\right|=\Delta$, then $|\operatorname{adj} A|=\Delta^{A-1}$
Here, adj $P_{3 \times 3}=$ $ \begin{bmatrix} 1 & 4 & 4\\ 2 & 1 & 7\\ 1 & 1 & 3 \end{bmatrix} $
$\Rightarrow \quad|\operatorname{adj} P|=|P|^{2}$
$ \begin{aligned} \therefore \quad|\operatorname{adj} P| & =\begin{bmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{bmatrix}=1(3-7)-4(6-7)+4(2-1) \\ & =-4+4+4=4 \Rightarrow|P|= \pm 2 \end{aligned} $
