Question

Identify the CORRECT statements for a $n \times n$ matrix.

• A. Under elementary row operations, the rank of the matrix remains invariant.
• B. Under elementary row operations, the eigenvalues of the matrix remain the same.
• C. If the elements in a row can be written as a linear combination of two or more rows, then the matrix is singular.
• D. The rank of the matrix is equal to $n$ if the determinant of the matrix is zero.

Hint:

A matrix is singular if its rows are linearly dependent, and its rank will be less than $n$.

Answer 1

The Correct Option is A, C

Step 1: Statement (A) Explanation.
Elementary row operations (like row swapping, scaling, or adding multiples of one row to another) do not change the rank of a matrix. Hence, statement (A) is correct.
Step 2: Statement (B) Explanation.
Elementary row operations do not preserve the eigenvalues of a matrix. While row operations affect the matrix's determinant, they do not necessarily maintain the eigenvalues. Therefore, statement (B) is incorrect.
Step 3: Statement (C) Explanation.
If the elements of a row can be expressed as a linear combination of two or more rows, the rows are linearly dependent, which makes the matrix singular (i.e., non-invertible). Hence, statement (C) is correct.
Step 4: Statement (D) Explanation.
If the determinant of a matrix is zero, it indicates that the rows are linearly dependent and the matrix is singular. However, the rank of the matrix is not necessarily equal to $n$ if the determinant is zero. If the determinant is zero, the matrix is singular, and its rank is less than $n$. Therefore, statement (D) is incorrect. Final Answer: If the elements in a row can be written as a linear combination of two or more rows, then the matrix is singular.