GATE 2022 | Maths | Q-12

Question

Consider

$P :$ Let \( M \in \mathbb{R}^{m \times n} \) with \( m > n \geq 2 \). If \( \text{rank}(M) = n \), then the system of linear equations \( Mx = 0 \) has \( x = 0 \) as the only solution.

$Q:$ Let \( E \in \mathbb{R}^{n \times n} \), \( n \geq 2 \), be a non-zero matrix such that \( E^3 = 0 \). Then \( I + E^2 \) is a singular matrix.

Which of the following statements is TRUE?

(A) Both P and Q are TRUE

(B) Both P and Q are FALSE

(C)P is TRUE and Q is FALSE

(D) P is FALSE and Q is TRUE