Matrix Statements Analysis

This question requires us to evaluate four statements about matrices and determine which ones are correct. Let's analyze each statement:

Statement A Analysis: Skew Symmetric Matrix Diagonal Elements

Statement A says: "In a skew symmetric matrix, all diagonal elements are zero."

• Definition: A square matrix $A$ is called skew symmetric if its transpose is equal to its negative, i.e., $A^T = -A$.
• Condition for Diagonal Elements: Let $A = [a_{ij}]$ be a matrix. For the transpose $A^T$, the element at the $i$-th row and $j$-th column is $a_{ji}$. The condition $A^T = -A$ means $a_{ji} = -a_{ij}$ for all $i$ and $j$.
• Diagonal Case ($i=j$): When we consider the diagonal elements, where the row index $i$ is equal to the column index $j$, the condition becomes $a_{ii} = -a_{ii}$.
• Solving for $a_{ii}$: Adding $a_{ii}$ to both sides gives $2a_{ii} = 0$. Dividing by 2, we get $a_{ii} = 0$.
• Conclusion: Therefore, all diagonal elements of a skew symmetric matrix must be zero. Statement A is correct.

Statement B Analysis: Diagonal Matrix Definition

Statement B says: "A square matrix is called a diagonal matrix, if all its non-diagonal elements are one."

• Definition: A square matrix is called a diagonal matrix if all its elements off the main diagonal are zero. The elements on the main diagonal can be any value, including zero.
• Comparison: The statement incorrectly states that non-diagonal elements must be 'one'. The correct condition is that they must be 'zero'.
• Conclusion: Statement B is incorrect.

Statement C Analysis: Singular Matrix Determinant

Statement C says: "If the determinant of the matrix is zero, then the matrix is known as non-singular matrix."

• Definitions:
  • A square matrix $A$ is called non-singular if its determinant is non-zero, i.e., $\det(A) \neq 0$.
  • A square matrix $A$ is called singular if its determinant is zero, i.e., $\det(A) = 0$.
• Comparison: The statement claims a matrix with a zero determinant is non-singular, which contradicts the definition. A matrix with a zero determinant is singular.
• Conclusion: Statement C is incorrect.

Statement D Analysis: Matrix Product with Adjoint

Statement D says: "The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinant A."

• Adjoint Property: A fundamental property relating a square matrix $A$ and its adjoint matrix, denoted as $\text{adj}(A)$, is given by the formula: $ A \cdot \text{adj}(A) = \det(A) \cdot I $ where $\det(A)$ is the determinant of matrix $A$, and $I$ is the identity matrix (also known as the unit matrix) of the same order as $A$.
• Explanation: The formula states that multiplying a matrix $A$ by its adjoint results in a matrix where each diagonal element is the determinant of $A$, and all off-diagonal elements are zero. This is equivalent to multiplying the identity matrix $I$ by the scalar value $\det(A)$.
• Conclusion: Statement D accurately describes this property. Statement D is correct.

Final Conclusion

Based on the analysis, statements A and D are correct, while statements B and C are incorrect.

Therefore, the correct option is the one that includes only statements A and D.