GATE CSE 1987 | Question: 1-xxiii

Question

A square matrix is singular whenever 

• A. The rows are linearly independent
• B. The columns are linearly independent
• C. The row are linearly dependent
• D. None of the above

Answer 1

When a row(or a column) is linearly dependent on some other rows(or columns) then it means that the particular row(or column) can be made by linear combination of other rows(or columns). So we can make this dependent row(or column) completely zero by subtracting from other rows(or columns) on which it is dependent. And so we have a zero row(or column) in the determinant and so determinant will be zero.

Hence, C is the correct option.

Answer 2

When the rows are linearly dependent the determinant of the matrix becomes $0$ hence in that case it will become singular matrix.

Hence, C is the correct option.

Comments

The determinant is zero, when the columns or rows of the matrix are linearly dependent.

Right????

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yes, for either rows or column

Answer 3

If the rows (or columns) of a square matrix are linearly dependent, then the determinant of matrix becomes zero.

Therefore, whenever the rows are linearly dependent, the matrix is singular.

Hence, C is the correct option.