Matrices & Determinants
3×3 matrix with four 1s — unique null vector
MJAT_TS6_P1
Grade 12
Question:
Let $P=[a_{ij}]_{3\times 3}$ with $\text{tr}(P)=2$ such that exactly four elements of $P$ are 1 and the rest are 0, and there exists a unique column matrix $M=(x,y,z)^T$ such that $PM=\mathbf{0}$. Then which is/are true?
A) Number of such matrices $P$ is 6
B) $\det(P)$ has two possible values
C) $\det(\text{adj}(P))$ can be $-1$
D) $\det(\text{adj}(P))$ can be $1$
Step-by-Step Solution
Key Concept: $\text{tr}(P)=2$ with four 1s: the diagonal has exactly two 1s, and two off-diagonal elements are 1. For $PM=0$ to have a unique (up to scalar) solution: $\text{rank}(P)=2\Rightarrow\det(P)=0$.
A ✓ (6 matrices), D ✓ ($\det(\text{adj}P)=1$ possible). Answer: A, D.
Correct Answer: AD