Solved: [M] Use a matrix program to compute the

Problem 2E Use row operations to show that the determinants in Exercises 2–4 are all zero. 12 13 14 15 16 17 18 19 20

Problem 20E [M] Use the method of Exercise 19 to guess the determinant of Justify your conjecture. [Hint: Use Exercise 14(c) and the result of Exercise 19.] Exercise 19: [M] Use a matrix program to compute the determinants of the following matrices. Use the results to guess the determinant of the matrix below, and confirm your guess by using row operations to evaluate that determinant. Exercise 14(c): Let A, B, C, D, and I be n × n matrices. Use the definition or properties of a determinant to justify the following formulas. Part (c) is useful in applications of eigenvalues (Chapter 5).

Problem 3E Use row operations to show that the determinants in Exercises 2–4 are all zero. 1 a b + c 1 b a + c 1 c a + b

Problem 1E Mark each statement True or False. Justify each answer. Assume that all matrices here are square. a. If A is a 2 × 2 matrix with a zero determinant, then one column of A is a multiple of the other. b. If two rows of a 3 × 3 matrix A are the same, then det A = 0. c. If A is a 3 × 3 matrix, then det 5A = 5 det A. d. If A and B are n × n matrices, with det A = 2 and det B = 3, then det.(A + B) = 5. e. If A is n × n and det A = 2, then det A3 = 6. f. If B is produced by interchanging two rows of A, then det B = det A. g. If B is produced by multiplying row 3 of A by 5, then det B = 5 · det A. h. If B is formed by adding to one row of A a linear combination of the other rows, then det B = det A. i. det j. det k. det l. Any system of n linear equations in n variables can be solved by Cramer’s rule. m. If u and v are in then the area of the triangle in the plane with vertices at 0, u, and v is 10. n. If = 0. o. If p. If

Problem 6E Compute the determinants in Exercises 5 and 6. 4 8 8 8 5 0 1 0 0 0 6 8 8 8 7 0 8 8 3 0 0 8 2 0 0

Problem 5E Compute the determinants in Exercises 5 and 6. 9 1 9 9 9 9 0 9 9 2 4 0 0 5 0 9 0 3 9 0 6 0 0 7 0

Problem 4E Use row operations to show that the determinants in Exercises 2–4 are all zero. a b c a + x b + x c + x a + y b + y c + y

Problem 8E Find a 3 × 3 determinant equation similar to that in Exercise 7 that describes the equation of the line through with slope m. Exercise 7 Show that the equation of the line in can be written as

Problem 7E Show that the equation of the line in can be written as

Problem 9E Exercises 9 and 10 concern determinants of the following Vandermonde matrices. Use row operations to show that

Problem 10E Exercises 9 and 10 concern determinants of the following Vandermonde matrices. Let all distinct. Explain why f (t) is a cubic polynomial, show that the coefficient of t3 is nonzero, and find three points on the graph of f

Problem 12E Use the concept of area of a parallelogram to write a statement about a 2 × 2 matrix A that is true if and only if A is invertible.

Problem 11E Determine the area of the parallelogram determined by the points How can you tell that the quadrilateral determined by the points is actually a parallelogram?

Problem 13E Show that if A is invertible, then adj A is invertible, and [Hint: Given matrices B and C, what calculation(s) would show that C is the inverse of B.]

Problem 14E Let A, B, C, D, and I be n × n matrices. Use the definition or properties of a determinant to justify the following formulas. Part (c) is useful in applications of eigenvalues (Chapter 5).

Problem 15E Let A, B, C, and D be n × n matrices with A invertible. a. Find matrices X and Y to produce the block LU factorization

Problem 16E Let J be the n × n matrix of all 1’s, and consider a. Subtract row 2 from row 1, row 3 from row 2, and so on, and explain why this does not change the determinant of the matrix. b. With the resulting matrix from part (a), add column 1 to column 2, then add this new column 2 to column 3, and so on, and explain why this does not change the determinant. c. Find the determinant of the resulting matrix from (b).

Problem 19E [M] Use a matrix program to compute the determinants of the following matrices. Use the results to guess the determinant of the matrix below, and confirm your guess by using row operations to evaluate that determinant.

Problem 17E Let A be the original matrix given in Exercise 16, and let Notice that A, B, and C are nearly the same except that the first column of A equals the sum of the first columns of B and C. A linearity property of the determinant function, discussed in Section 3.2, says that det A = det B + det C. Use this fact to prove the formula in Exercise 16 by induction on the size of matrix A. Exercise 16: Let J be the n × n matrix of all 1’s, and consider a. Subtract row 2 from row 1, row 3 from row 2, and so on, and explain why this does not change the determinant of the matrix. b. With the resulting matrix from part (a), add column 1 to column 2, then add this new column 2 to column 3, and so on, and explain why this does not change the determinant. c. Find the determinant of the resulting matrix from (b).

Problem 18E [M] Apply the result of Exercise 16 to find the determinants of the following matrices, and confirm your answers using a matrix program. Exercise 16: Let J be the n × n matrix of all 1’s, and consider a. Subtract row 2 from row 1, row 3 from row 2, and so on, and explain why this does not change the determinant of the matrix. b. With the resulting matrix from part (a), add column 1 to column 2, then add this new column 2 to column 3, and so on, and explain why this does not change the determinant. c. Find the determinant of the resulting matrix from (b).