1. Prove that the determinant of an upper triangular matrix is the product of its diagonal elements. 2. Prove that if a row of a square matrix  is multiplied by a scalar  then the determinant of the...

1. Prove that the determinant of an upper triangular matrix is the product of its diagonal elements.

2. Prove that if a row of a square matrix
 is multiplied by a scalar
 then the determinant of the modified matrix is
 det

3. In this problem, you will prove that if two rows of a matrix are interchanged, the determinant changes sign. Represent the matrix as a column of rows:

By using this representation, explain the validity of each step in the proof.

 Add row
 to row

 Subtract row
 from row

 Add row j to row

 Multiply row
 by
 and the proof is complete.