Question

Consider the following statements related to a matrix:
A. Inverse of a matrix is unique if it exists.
B. A matrix \(A\) is non-singular if \(|A|=0\).
C. Matrix \[ A=\begin{pmatrix} \cos x & \sin x -\sin x & \cos x \end{pmatrix} \] is orthogonal.
D. Matrix \[ A=\begin{pmatrix} 0 & -2 & -8 2 & 0 & -4 8 & -4 & 0 \end{pmatrix} \] is skew symmetri
C.

• A. A, B, C Only
• B. A, C, D Only
• C. A, C Only
• D. A, D Only

Hint:

A non-singular matrix has non-zero determinant, and an orthogonal matrix satisfies \(AA^T=I\).

Answer 1

The Correct Option is C

Concept:
A matrix inverse, orthogonal matrix and skew-symmetric matrix have fixed standard properties.

Step 1: Check statement A.
If inverse of a matrix exists, then it is unique. \[ A \text{ is correct} \]

Step 2: Check statement B.
A matrix is non-singular when: \[ |A|\neq 0 \] But statement B says: \[ |A|=0 \] So B is incorrect.

Step 3: Check statement C.
For: \[ A=\begin{pmatrix} \cos x & \sin x -\sin x & \cos x \end{pmatrix} \] we get: \[ AA^T=I \] So the matrix is orthogonal. \[ C \text{ is correct} \]

Step 4: Check statement D.
For a skew-symmetric matrix: \[ a_{ij}=-a_{ji} \] Here: \[ a_{23}=-4,\quad a_{32}=-4 \] But for skew-symmetric matrix, these should be opposite in sign. So D is incorrect. Therefore, correct statements are: \[ A,C \] \[ \therefore \text{Correct Answer is (C)} \]