Discussion
Determinant of a matrix remains unaltered if ______.
Answer: A
The determinant is unaltered if its columns and rows are interchanged, that is, on transposition: \(\det(A^{T}) = \det(A)\).
Here a “line” means a row or a column. Expanding along the first row of \(A\) gives the same terms as expanding along the first column of \(A^{T}\), which is why every property of rows also holds for columns.
The other operations change the determinant. If two parallel lines (two rows or two columns) are identical, the determinant becomes zero. Parallel lines of a matrix are separate rows or columns and cannot intersect, so that option describes nothing. Multiplying every element of one line by a factor \(k\) multiplies the determinant by \(k\).
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