Question

Which of the following statements are correct?
A. In a skew-symmetric matrix, all diagonal elements are zero.
B. A square matrix is called a diagonal matrix if all its non-diagonal elements are one.
C. If the determinant of the matrix is zero, then the matrix is known as non-singular matrix.
D. The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinant A.

• A. A and D only
• B. B and C only
• C. A, B and D only
• D. C and D only

Hint:

Memorize the fundamental definitions and properties of matrices: - Skew-symmetric: \(A^T = -A\) - Singular: \(\det(A) = 0\) - Adjoint property: \(A \cdot \text{adj}(A) = \det(A) \cdot I\)

Answer 1

The Correct Option is A

Step 1: Analyze statement A.
A matrix M is skew-symmetric if \( M^T = -M \), which means \( m_{ji} = -m_{ij} \) for all i, j. For diagonal elements, \(i=j\), so \( m_{ii} = -m_{ii} \). This implies \( 2m_{ii} = 0 \), so \( m_{ii} = 0 \). Thus, all diagonal elements of a skew-symmetric matrix are zero. Statement A is correct.

Step 2: Analyze statement B.
A square matrix is a diagonal matrix if all of its non-diagonal elements are zero. The statement says they are one, which is incorrect. Statement B is incorrect.

Step 3: Analyze statement C.
A matrix is called singular if its determinant is zero. It is called non-singular if its determinant is non-zero. The statement claims the opposite. Statement C is incorrect.

Step 4: Analyze statement D.
A fundamental property of matrices states that for any square matrix A, \( A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I \), where I is the identity (unit) matrix. The statement says "unit matrix multiplied by the determinant A", which is exactly this property. Statement D is correct. Conclusion: Statements A and D are correct.