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

Hint:

For a matrix to have a full rank, the number of linearly independent columns must equal the number of columns. Also, a nilpotent matrix raised to some power results in the zero matrix.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement P: The statement in P is true. If the rank of the matrix \( M \) is \( n \), the system \( Mx = 0 \) will only have the trivial solution \( x = 0 \). This follows from the fact that if a matrix has full column rank (i.e., rank = number of columns), then the null space contains only the zero vector. 
Step 2: Analyzing Statement Q: The statement in Q is false. It is given that \( E^3 = 0 \), meaning that \( E \) is a nilpotent matrix. For \( I + E^2 \) to be singular, \( I + E^2 \) must have a determinant of zero. However, \( I + E^2 \) is not singular because \( E^2 \) is a nilpotent matrix, and adding the identity matrix \( I \) ensures that the resulting matrix is non-singular. Hence, statement Q is false. 
Thus, the correct answer is (C) P is TRUE and Q is FALSE.