Question

Consider the following statements:
(A). If AB = 0, it implies either A or B is a null matrix
(B). If determinant of a square matrix is non-zero, then it is non-singular
(C). Adjoint of symmetric matrix is symmetric
(D). Adjoint of a diagonal matrix is diagonal
Which of the above statements are true

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

Hint:

Always keep in mind that matrix multiplication is non-commutative (\(AB \neq BA\)) and permits zero divisors. These two distinct differences from scalar algebra are frequently tested in competitive engineering exams.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem tests basic properties of square matrices, including zero divisors, the definition of non-singularity, and the characteristics of the adjoint matrix when applied to symmetric and diagonal matrices.

Step 2: Detailed Explanation:
Let us analyze each statement individually:

Statement (A): In matrix algebra, unlike real number arithmetic, the product of two non-zero (non-null) matrices can result in a null matrix. These are referred to as divisors of zero.
For instance, consider: \[ A = \begin{pmatrix} 1 & 0 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 0 0 & 1 \end{pmatrix} \] Neither \(A\) nor \(B\) is a null matrix, yet their product is: \[ AB = \begin{pmatrix} 1\times0 + 0\times0 & 1\times0 + 0\times1 0\times0 + 0\times0 & 0\times0 + 0\times1 \end{pmatrix} = \begin{pmatrix} 0 & 0 0 & 0 \end{pmatrix} = 0 \] Therefore, \(AB = 0\) does not imply that either \(A\) or \(B\) must be a null matrix. Statement (A) isfalse.

Statement (B): A square matrix \(M\) is defined as non-singular if and only if its determinant is non-zero, i.e., \(\det(M) \neq 0\). This condition is also the prerequisite for the matrix to have a unique inverse. Statement (B) istrue.

Statement (C): A matrix \(A\) is symmetric if \(A^T = A\).
For any square matrix, we have the identity: \[ (\text{adj}(A))^T = \text{adj}(A^T) \] Substituting \(A^T = A\) into this identity yields: \[ (\text{adj}(A))^T = \text{adj}(A) \] This indicates that the transpose of the adjoint of a symmetric matrix is equal to the adjoint itself, which is the definition of a symmetric matrix. Statement (C) istrue.

Statement (D): A diagonal matrix has non-zero elements only along its principal diagonal, meaning its off-diagonal elements are all zero.
The cofactor of any off-diagonal element \(a_{ij}\) (where \(i \neq j\)) in a diagonal matrix always contains at least one row or column of all zeros, making its determinant zero.
Consequently, the cofactors of all off-diagonal elements are zero, which ensures that the adjoint matrix remains diagonal. Statement (D) istrue.
Thus, statements (B), (C), and (D) are true, while statement (A) is false.

Step 3: Final Answer:
The true statements are (B), (C), and (D), which corresponds to Option (B).