Question

Let \(a=\begin{bmatrix} \frac{1}{\sqrt3} \\ \frac{-1}{\sqrt2} \\ \frac{1}{\sqrt6} \\ 0\end{bmatrix}\). Consider the following two statements.
P : The matrix I₄ - aa^T is invertible.
Q: The matrix I₄ - 2aa^T is invertible.
Then, which one of the following holds ?

• A. P is false but Q is true
• B. P is true but Q is false
• C. Both P and Q are true
• D. Both P and Q are false

Answer 1

The Correct Option is A

- Statement P: The matrix \( I_4 - aa^T \) is not invertible. To understand this, note that \( aa^T \) is a rank-1 matrix, meaning that its determinant is 0. Therefore, the matrix \( I_4 - aa^T \) has a rank of 3 (since it can have at most 3 independent rows). This implies that the matrix is singular, and hence not invertible. Therefore, P is false. - Statement Q: The matrix \( I_4 - 2aa^T \) is invertible. To check this, we calculate the rank of the matrix. The matrix \( 2aa^T \) is also a rank-1 matrix, and we subtract it from the identity matrix. The subtraction of a rank-1 matrix from a rank-4 matrix (the identity matrix) results in a matrix of full rank (rank 4). Hence, \( I_4 - 2aa^T \) is invertible. Therefore, Q is true. Thus, the correct answer is (A): P is false but Q is true.