GATE 2026 DA – Question 52

Linear Algebra · Special matrices: projection, orthogonal and idempotent, quadratic forms · 2 marks · Multiple select

Let $M = \left(I_n - \frac{1}{n}\mathbf{1}\mathbf{1}^T\right)$ be a matrix, where $\mathbf{1} = (1, 1, 1, \ldots, 1)^T \in \mathbb{R}^n$ and $I_n$ is the identity matrix of order $n$.

Which of the following options is/are correct?

$M^T = M$
$M^2 = I_n$
$Trace(M) = n$
$M$ is a projection matrix

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Show answer and explanation

Correct answer: (A) $M^T = M$; (D) $M$ is a projection matrix

Explanation

$\mathbf{1}\mathbf{1}^T$ is symmetric, so $M^T = M$ (A is true). Since $\mathbf{1}^T\mathbf{1} = n$, $\left(\frac{1}{n}\mathbf{1}\mathbf{1}^T\right)^2 = \frac{1}{n}\mathbf{1}\mathbf{1}^T$, which makes $M^2 = M$ (it is idempotent, and not equal to $I_n$, so B is false). A symmetric idempotent matrix is an orthogonal projection (onto the vectors perpendicular to $\mathbf{1}$), so D is true. The trace is $n - \frac{1}{n}\cdot n = n - 1$, so C is false.