Question

Determinant of a matrix remains unaltered if __________.

• A. its columns and rows are interchanged
• B. two parallel lines are identical
• C. two parallel lines intersect
• D. each element of a line is multiplied by the same factor

Hint:

Interchanging rows or columns only changes the sign of the determinant. However, multiplying a row or column by a scalar will multiply the determinant by that scalar.

Answer 1

The Correct Option is A

The determinant of a matrix is a special number calculated from the elements of the matrix. It has several important properties related to matrix transformations. Let’s analyze the options: - Option (A): "its columns and rows are interchanged" - This is correct. Interchanging two rows or two columns of a matrix results in a change in the sign of the determinant. However, this does not change its absolute value, meaning the determinant remains "unaltered" in magnitude, only the sign changes. - Option (B): "two parallel lines are identical" - This is incorrect. When two rows or columns of a matrix are identical, the determinant becomes zero. This is because the matrix is singular (its rows or columns are linearly dependent). - Option (C): "two parallel lines intersect" - This is correct. When two lines (or rows/columns) are identical, the determinant becomes zero, which is a special case of linear dependence. However, this implies that the determinant is zero, which is consistent with the definition of the determinant. - Option (D): "each element of a line is multiplied by the same factor" - This is incorrect. When all elements of a row or column of a matrix are multiplied by a scalar, the determinant is multiplied by that scalar. Therefore, the determinant changes and does not remain the same. Therefore, the correct answers are (A) and (C), as these are the cases that involve no change in the absolute value of the determinant.