In , evaluate each given determinant after

Problem 4P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 1P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 2P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 5P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 3P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 64P Problem, deal with the Vandermonde determinant Use the formula to evaluate the two determinants given next. Formula

Problem 6P Use cofactor expansions to evaluate the determinants in Problem. Expand along the row or column that minimizes the amount of computation that is required.

Problem 7P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 8P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 9P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 10P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 11P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 12P In Problem, evaluate each given determinant after first simplifying the computation (as in Example 6) by adding an appropriate multiple of some row or column to another.

Problem 13P Use the method of elimination to evaluate the determinants in Problem

Problem 15P Use the method of elimination to evaluate the determinants in Problem

Problem 14P Use the method of elimination to evaluate the determinants in Problem

Problem 16P Use the method of elimination to evaluate the determinants in Problem

Problem 17P Use the method of elimination to evaluate the determinants in Problem

Problem 18P Use the method of elimination to evaluate the determinants in Problem

Problem 19P Use the method of elimination to evaluate the determinants in Problem

Problem 20P Use the method of elimination to evaluate the determinants in Problem

Problem 21P Use Cramer’s rule to solve the systems in Problem. 3x + 4y = 2 5x + 7y = 1

Problem 22P Use Cramer’s rule to solve the systems in Problem. 5x + 8y = 38x + 13y = 5

Problem 23P Use Cramer’s rule to solve the systems in Problem. 17x + 7y = 612x + 5y = 4

Problem 24P Use Cramer’s rule to solve the systems in Problem. 11x + 15y = 108x + 11y = 7

Problem 25P Use Cramer’s rule to solve the systems in Problem. 5x + 6y = 123x + 4y = 6

Problem 26P Use Cramer’s rule to solve the systems in Problem. 6x + 7y = 38x + 9y = 4

Problem 27P Use Cramer’s rule to solve the systems in Problem. 5x1 + 2x2 ? 2x3 = 1x1 + 5x2 ? 3x3 = ?2 5x1 ? 3x2 + 5x3 = 2

Problem 30P Use Cramer’s rule to solve the systems in Problem. x1 + 4x2 + 2x3 = 34x1 + 2x2 + x3 = 1 2x1 ? 2x2 ? 5x3 = ?3

Problem 28P Use Cramer’s rule to solve the systems in Problem. 5x1 + 4x2 ? 2x3 = 42x1 + 3x3 = 2 2x1 ? x2 + x3 = 1

Problem 29P Use Cramer’s rule to solve the systems in Problem.

Problem 31P Use Cramer’s rule to solve the systems in Problem. 2x1 ? 5x3 = ?34x1 ? 5x2 + 3x3 = 3 ?2x1 + x2 + x3 = 1

Problem 32P Use Cramer’s rule to solve the systems in Problem. 3x1 + 4x2 ? 3x3 = 53x1 ? 2x2 + 4x3 = 7 3x1 + 2x2 ? x3 = 3

Problem 35P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 37P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 34P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 36P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 38P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 33P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem .

Problem 39P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 40P Apply Theorem 5 to find the inverse A?1 of each matrix A given in Problem

Problem 41P Show that (AB) T = BT AT if A and B are arbitrary 2 × 2 matrices.

Problem 43P In the Problem lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column.

Problem 44P In the Problem lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column.

Problem 45P In the Problem lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column.

Problem 42P Consider the 2 × 2 matrices and where x and y denote the row vectors of B. Then the product AB can be written in the form Use this expression and the properties of determinants to show that Thus the determinant of a product of 2 × 2 matrices is equal to the product of their determinants.

Problem 46P In the Problem lists a special case of one of Property 1 through Property 5. Verify it by expanding the determinant on the left-hand side along an appropriate row or column.

Problem 49P Let A = [ aij ] be a 3 × 3 matrix. Show that det(AT) = det A by expanding det A along its first row and det(AT) along its first column.

Problem 47P Suppose that A and B are matrices of the same size. Show that: (a) (AT) T = A; ________________ (b) (cA) T = cAT; and ________________ (c) (A + B) T = AT + BT .

Problem 50P Suppose that A2 = A. Prove that |A| = 0 or |A| = 1.

Problem 51P Suppose that An = 0 (the zero matrix) for some positive integer n. Prove that |A| = 0.

Problem 52P The square matrix A is called orthogonal provided that AT = A?1. Show that the determinant of such a matrix must be either + 1 or ? 1.

Problem 53P The matrices A and B are said to be similar provided that A = P?1BP for some invertible matrix P. Show that if A and B are similar, then |A| = |B|.

Problem 54P Deduce from Theorems 2 and 3 that if A and B are a n × n invertible matrices, then AB is invertible if and only if both A and B are invertible.

Problem 56P Let A be an n × n matrix with det A = 1 and with all elements of A integers. (a) Show that A?1 has only integer entries. ________________ (b) Suppose that b is an n-vector with only integer entries. Show that the solution vector x of Ax= b has only integer entries. Let A be a 3 × 3 upper triangular matrix with nonzero determinant. Show by explicit computation that A?1 is also upper triangular.

Problem 55P Let A and B be n × n matrices. Suppose it is known that either AB = I or BA = I. Use the result of Problem 1 to conclude that B = A?1. Problem 1 Deduce from Theorems 2 and 3 that if A and B are a n × n invertible matrices, then AB is invertible if and only if both A and B are invertible.

Problem 59P Show that and

Problem 61P Problem, deal with the Vandermonde determinant Show by direct computation that V(a, b) = b ? a and that