A and B are n times n matrices Check the true statements bel

A| and B| are n times n matrices. Check the true statements below: A. if the columns of A| are linearly dependent, then det A = 0. B. Adding a multiple of one row to another does not affect the determinant of a matrix. C. det (A+B) = det A + det B|. D. The determinant of A| is the product of the pivots in any echelon form U| of A|, multiplied by (-1)^tau|, where r is the number of row interchanges made during row reduction from A| to U|.

Solution

A True, if the columns are linearly dependent the matrix is singular so det A = 0

B True, you can prove it by expanding the deteminant about the row that you changed. You get the sum of the original determinant + the determinant of a singular matrix (which is 0).

C False.This is true for product however.

D FALSE If we scale any rows when getting the echelon form, we change the determinant.