4.6. EXERCISES 99

38. Show that 4.20 is valid for p = −1 if and only if each block has an inverse and thatthis condition holds if and only if A is invertible.

39. Let A be an n×n matrix and let Pi j be the permutation matrix which switches the ith

and jth rows of the identity. Show that Pi jAPi j produces a matrix which is similar toA which switches the ith and jth entries on the main diagonal.

40. 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.

41. Consider the symmetric 3× 3 matrices, those for which A = AT . Show that withrespect to the usual notions of addition and scalar multiplication this is a vector spaceof dimension 6. What is the dimension of the set of skew symmetric matrices?

42. You have an m×n matrix of rank r. Explain why if you delete a column, the resultingmatrix has rank r or rank r−1.

43. Using the fact that multiplication on the left by an elementary matrix accomplishes arow operation, show easily that row operations produce no change in linear relationsbetween columns.