Discussion
Consider
P: Let \(M \in \mathbb{R}^{m \times n}\) with \(m \gt n \geq 2\). If 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?
Answer: C
The answer is P is TRUE and Q is FALSE.
P: rank(M) = n means the n columns of M are linearly independent. Mx is a combination of the columns with weights from x, so Mx = 0 forces every weight to be zero, that is x = 0. P is true; m \(\gt\) n only makes the matrix tall.
Q: from \(E^{3} = 0\) we get \(E^{4} = 0\), so
\[ (I + E^{2})(I - E^{2}) = I - E^{4} = I \]
Thus \(I + E^{2}\) has the inverse \(I - E^{2}\) and is non-singular, so Q is false. In general, I plus a nilpotent matrix is always invertible. That rules out every option calling Q true.
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