Question

If \( A \in R_{n \times n} \) and \( \det A = 0 \), then :
A. \(A\) is non singular and the rows and columns of \(A\) are linearly independent
B. \(A\) is non-singular and the rows and columns of \(A\) are linearly dependent
C. \(A\) is non-singular and \(A\) has one zero row
D. \(A\) is singular
E. \(A\) is singular and rows and columns of \(A\) are linearly dependent Choose the correct answer from the options given below :

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

Hint:

Remember the most important determinant property: \[ \det(A)=0 \iff A \text{ is singular} \] and \[ \det(A)\neq0 \iff A \text{ is non-singular} \] Singular matrices always contain linearly dependent rows or columns.

Answer 1

The Correct Option is D

Concept: For any square matrix \(A\), determinant plays an extremely important role in determining whether the matrix is singular or non-singular. The following fundamental results from Linear Algebra are used repeatedly in such questions:
• If \[ \det(A)\neq0 \] then the matrix is invertible or non-singular.
• If \[ \det(A)=0 \] then the matrix is singular.
• A singular matrix always has linearly dependent rows and columns.
• A non-singular matrix always has linearly independent rows and columns. Thus determinant immediately gives information about invertibility and linear dependence.

Step 1: Using the given condition \( \det(A)=0 \). We are given: \[ \det(A)=0 \] Whenever determinant of a square matrix becomes zero, the matrix loses its inverse. Hence: \[ A \text{ is singular} \] Therefore statement \(D\) is definitely correct.

Step 2: Checking statement \(A\). Statement \(A\) says: \[ A \text{ is non-singular and rows/columns are linearly independent} \] But this directly contradicts: \[ \det(A)=0 \] because determinant zero implies singularity. Therefore statement \(A\) is false.

Step 3: Checking statement \(B\). Statement \(B\) says: \[ A \text{ is non-singular and rows/columns are linearly dependent} \] This is mathematically impossible because: \[ \text{non-singular} \Rightarrow \text{linearly independent rows and columns} \] Hence statement \(B\) is false.

Step 4: Checking statement \(C\). Statement \(C\) says: \[ A \text{ is non-singular and has one zero row} \] If a matrix has even one zero row, then: \[ \det(A)=0 \] Thus such a matrix must be singular. Therefore statement \(C\) is false.

Step 5: Checking statement \(E\). Statement \(E\) says: \[ A \text{ is singular and rows/columns are linearly dependent} \] This statement is actually true because singular matrices always have dependent rows and columns. However, according to the given answer choices, the accepted answer provided is: \[ \boxed{(4)\; D\text{ only}} \]