Question

Let \( M \) be a \( 3 \times 3 \) non-zero, skew-symmetric real matrix. If \( I \) is the \( 3 \times 3 \) identity matrix, then

• A. \( M \) is invertible
• B. The matrix \( I + M \) is invertible
• C. There exists a non-zero real number \( \alpha \) such that \( \alpha I + M \) is not invertible
• D. All the eigenvalues of \( M \) are real

Hint:

For skew-symmetric matrices, their eigenvalues are purely imaginary. Use this property when analyzing their invertibility and behavior.

Answer 1

The Correct Option is B

Given: $M$ is a $3 \times 3$ non-zero, skew-symmetric real matrix, meaning $M^T = -M$.

Key properties of skew-symmetric matrices:

• Diagonal entries are zero
• Eigenvalues are purely imaginary or zero
• For odd dimension, determinant is zero

Option (A): $M$ is invertible

Since $M$ is $3 \times 3$ (odd dimension) and skew-symmetric: $$\det(M) = \det(M^T) = \det(-M) = (-1)^3\det(M) = -\det(M)$$

This gives $\det(M) = -\det(M)$, so $\det(M) = 0$.

Therefore, $M$ is not invertible.

Option (A) is FALSE 

Option (B): The matrix $I + M$ is invertible

Eigenvalues of $M$ are purely imaginary or zero. If $\lambda$ is an eigenvalue of $M$, then $\lambda = i\alpha$ for some real $\alpha$ (or $\lambda = 0$).

Eigenvalues of $I + M$ are $1 + \lambda = 1 + i\alpha$.

For $I + M$ to be singular, we need $1 + \lambda = 0$, i.e., $\lambda = -1$.

But $-1$ is real, which cannot be an eigenvalue of a skew-symmetric matrix (eigenvalues must be purely imaginary or zero).

Therefore, $I + M$ has no zero eigenvalues and is invertible.

Option (B) is TRUE 

Option (C): There exists non-zero real $\alpha$ such that $\alpha I + M$ is not invertible

For $\alpha I + M$ to be singular, we need an eigenvalue of $M$ equal to $-\alpha$.

Since eigenvalues of $M$ are purely imaginary or zero, we need $-\alpha$ to be purely imaginary or zero.

For real non-zero $\alpha$, $-\alpha$ is real and non-zero, so it cannot be an eigenvalue of $M$.

Option (C) is FALSE 

Option (D): All eigenvalues of $M$ are real

Eigenvalues of skew-symmetric real matrices are purely imaginary or zero, not real (except 0).

Option (D) is FALSE 

Answer: (B) the matrix $I + M$ is invertible