128 CHAPTER 4. ROW OPERATIONS

30. Let A be an n×n matrix and let P ij be the permutation matrix which switches the ith

and jth rows of the identity. Show that P ijAP ij produces a matrix which is similarto A which switches the ith and jth entries on the main diagonal.

31. Recall the procedure for finding the inverse of a matrix on Page 49. It was shown thatthe procedure, when it works, finds the inverse of the matrix. Show that wheneverthe matrix has an inverse, the procedure works.

32. If EA = B where E is invertible, show that A and B have the same linear relationshipsamong their columns.

33. You could define column operations by analogy to row operations. That is, youswitch two columns, multiply a column by a nonzero scalar, or add a scalar multipleof a column to another column. Let E be one of these column operations applied tothe identity matrix. Show that AE produces the column operation on A which wasused to define E.