Solved: Explain why the columns of an n × n matrix A span

Problem 1E Find the inverses of the matrices in Exercises 1–4.

Problem 3E Find the inverses of the matrices in Exercises 1–4.

Problem 4E Find the inverses of the matrices in Exercises 1–4.

Problem 2E Find the inverses of the matrices in Exercises 1–4.

Problem 42E [M] With D as in Exercise 41, determine the forces that produce a deflection of .22 cm at the second point on the beam, with zero deflections at the other three points. How is the answer related to the entries in D–1? [Hint: First answer the question when the deflection is 1 cm at the second point.] Exercise 41: [M] Let be a flexibility matrix for an elastic beam such as the one in Example 3, with four points at which force is applied. Units are centimeters per newton of force. Measurements at the four points show deflections of .07, .12, .16, and .12 cm. Determine the forces at the four points. Example 3:

Problem 6E Use the inverse found in Exercise 3 to solve the system Exercise 3: Find the inverses of the matrices.

Problem 5E Use the inverse found in Exercise 1 to solve the system Exercise 1: Find the inverses of the matrices.

Problem 7E a. Find A–1 and use it to solve the four equations b. The four equations in part (a) can be solved by the same set of row operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix.

Problem 8E Suppose P is invertible and A = PBP–1 Solve for B in terms of A.

Problem 11E Let A be an invertible n × n matrix, and let B be an n × p matrix. Show that the equation AX = B has a unique solution A–1 B.

Problem 10E In Exercises 9 and 10, mark each statement True or False. Justify each answer. a. If A is invertible, then elementary row operations that reduce A to the identity In also reduce A–1 to In b. If A is invertible, then the inverse of A–1 is A itself. c. A product of invertible n × n matrices is invertible, and the inverse of the product is the product of their inverses in the same order. d. If A is an n × n matrix and Ax = ej is consistent for every then A is invertible. Note: e1,…, en represent the columns of the identity matrix. e. If A can be row reduced to the identity matrix, then A must be invertible.

Problem 12E Use matrix algebra to show that if A is invertible and D satisfies AD = I , then D = A–1

Problem 13E Suppose AB = AC, where B and C are n × p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible?

Problem 14E Suppose (B – C)D = 0 where B and C are m × n matrices and D is invertible. Show that B = C.

Problem 9E In Exercises 9 and 10, mark each statement True or False. Justify each answer. a. In order for a matrix B to be the inverse of A, the equations AB = I and BA = I must both be true. b. If A and B are n × n and invertible, then A–1B–1 is the inverse of AB. c. If then A is invertible. d. If A is an invertible n × n matrix, then the equation Ax = b is consistent for each b in ?n. e. Each elementary matrix is invertible.

Problem 16E Suppose A and B are n × n matrices, B is invertible, and AB is invertible. Show that A is invertible. [Hint: Let C = AB, and solve this equation for A.]

Problem 15E Let A be an invertible n × n matrix, and let B be an n × p matrix. Explain why A–1 B can be computed by row reduction: If If A is larger than 2 × 2; then row reduction of [A B] is much faster than computing both A–1 and A–1 B.

Problem 17E Suppose A, B, and C are invertible n × n matrices. Show that ABC is also invertible by producing a matrix D such that

Problem 18E Solve the equation AB = BC for A, assuming that A, B, and C are square and B is invertible.

Problem 19E If A, B and C are n × n invertible matrices, does the equation have a solution, X? If so, find it.

Problem 20E Suppose A, B, and X are n × n matrices with A, X, and A – AX invertible, and suppose a. Explain why B is invertible. b. Solve equation (3) for X. If a matrix needs to be inverted, explain why that matrix is invertible.

Problem 21E Explain why the columns of an n × n matrix A are linearly independent when A is invertible.

Problem 22E Explain why the columns of an n × n matrix A span ?n when A is invertible. [Hint: Review Theorem 4 in Section 1.4.] Theorem 4:

Problem 24E Suppose A is n × n and the equation Ax = b has a solution for each b in ?n. Explain why A must be invertible. [Hint: Is A row equivalent to In].

Problem 26E Exercises 25 and 26 prove Theorem 4 for Show that if ad – bc ? 0, the formula for A–1 works.

Problem 25E Exercises 25 and 26 prove Theorem 4 for Show that if ad – bc = 0, then the equation Ax = 0 has more than one solution. Why does this imply that A is not invertible? [Hint: First, consider a = b = 0: Then, if a and b are not both zero, consider the vector

Problem 23E Suppose A is n × n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In. By Theorem 7, this shows that A must be invertible. (This exercise and Exercise 24 will be cited in Section 2.3.) Theorem 7: An n × n matrix A is invertible if and only if A is row equivalent to In, and in this case, any sequence of elementary row operations that reduces A to In also transforms In into A–1.

Problem 27E Exercises 27 and 28 prove special cases of the facts about elementary matrices stated in the box following Example 5. Here A is a 3 × 3 matrix and I = I3. (A general proof would require slightly more notation.) Let A be a 3 × 3 matrix. a. Use equation (2) from Section 2.1 to show that rowi. (A) = rowi. (I)·A i = 1, 2, 3. b. Show that if rows 1 and 2 of A are interchanged, then the result may be written as EA, where E is an elementary matrix formed by interchanging rows 1 and 2 of I. c. Show that if row 3 of A is multiplied by 5, then the result may be written as EA, where E is formed by multiplying row 3 of I by 5. Example 5: Let Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A.

Problem 28E Exercises 27 and 28 prove special cases of the facts about elementary matrices stated in the box following Example 5. Here A is a 3 × 3 matrix and I = I3. (A general proof would require slightly more notation.) Suppose row 2 of A is replaced by row2. (A) – 3. row1 (A). Show that the result is EA, where E is formed from I by replacing row2 (I) by row2. (I) – 3 row1 (A) Example 5: Let Compute E1A, E2A, and E3A, and describe how these products can be obtained by elementary row operations on A.

Problem 29E Find the inverses of the matrices in Exercises 29–32, if they exist. Use the algorithm introduced in this section.

Problem 31E Find the inverses of the matrices in Exercises 29–32, if they exist. Use the algorithm introduced in this section.

Problem 32E Find the inverses of the matrices in Exercises 29–32, if they exist. Use the algorithm introduced in this section.

Problem 30E Find the inverses of the matrices in Exercises 29–32, if they exist. Use the algorithm introduced in this section.

Problem 33E Use the algorithm from this section to find the inverses of A be the corresponding n × n matrix, and let B be its inverse. Guess the form of B, and then show that AB = I .

Problem 34E Repeat the strategy of Exercise 33 to guess the inverse B of Show that AB = I Exercise 33: Use the algorithm from this section to find the inverses of A be the corresponding n × n matrix, and let B be its inverse. Guess the form of B, and then show that AB = I .

Problem 35E Let Find the third column of A–1 without computing the other columns.

Problem 36E [M] Let Find the second and third columns of A–1 without computing the first column

Problem 37E Let Construct a 2 × 3 matrix C (by trial and error) using only 1, –1, and 0 as entries, such that CA = I2. Compute AC and note that AC ? I3:

Problem 38E Let Construct a 4 ? 2 matrix D using only 1 and 0 as entries, such that AD = I2.Is it possible that CA = I4 for some 4 × 2 matrix C? Why or why not?

Problem 39E [M] Let be a flexibility matrix, with flexibility measured in inches per pound. Suppose that forces of 40, 50, and 30 lb are applied at points 1, 2, and 3, respectively, in Fig. 1 of Example 3. Find the corresponding deflections. Fig. 1 of Example 3:

Problem 40E [M] Compute the stiffness matrix D–1 for D in Exercise 39. List the forces needed to produce a deflection of .04 in. at point 3, with zero deflections at the other points. Exercise 39: [M] Let be a flexibility matrix, with flexibility measured in inches per pound. Suppose that forces of 40, 50, and 30 lb are applied at points 1, 2, and 3, respectively, in Fig. 1 of Example 3. Find the corresponding deflections. Fig. 1 of Example 3:

Problem 41E [M] Let be a flexibility matrix for an elastic beam such as the one in Example 3, with four points at which force is applied. Units are centimeters per newton of force. Measurements at the four points show deflections of .07, .12, .16, and .12 cm. Determine the forces at the four points. Example 3: