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: Evaluate statement A.
For a skew-symmetric matrix M, \( M^T = -M \), implying \( m_{ji} = -m_{ij} \) for all i and j. When \(i=j\), \( m_{ii} = -m_{ii} \), thus \( 2m_{ii} = 0 \) and \( m_{ii} = 0 \). Therefore, statement A, asserting that all diagonal elements of a skew-symmetric matrix are zero, is correct.

Step 2: Evaluate statement B.
A diagonal matrix has all non-diagonal elements equal to zero. Statement B incorrectly claims these elements are one. Thus, statement B is incorrect.

Step 3: Evaluate statement C.
A singular matrix has a determinant of zero, while a non-singular matrix has a non-zero determinant. Statement C incorrectly reverses this definition, and is therefore incorrect.

Step 4: Evaluate statement D.
For any square matrix A, \( A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I \), where I is the identity matrix. Statement D accurately reflects this property as "unit matrix multiplied by the determinant A". Thus, statement D is correct.Conclusion: Statements A and D are correct.