33,37 ix Algebra~1/ X], then X = A¹B... then row reduction of [A B] is muchboth A¹ and AB.here B and Caren xp matrices and AB = C. Is this true, in general, when0, where B and Care in x # matricesw that B = C.invertible x n matrices. Show thatby producing a matrix D such thatABC) = 1.xn, B is invertible, and AB is invert-rtible. [Hinr: Let C = AB, and solve= BC for A, assuming that A, B, andvertible.and A = PBP. Solve for B innvertible matrices, does the equationhave a solution, X? If so, find it.rex matrices with A, X, andsupposeertible.u need to invert a matrix, explainertible.of an n x n matrix A are linearlyvertible.(3)anx matrix A span R" wheniew Theorem 4 in Section 1.4.]he equation Ax = 0 has only theby A has n pivot columns and A isheorem 7, this shows that A muste and Exercise 24 will be cited ine equation Ax=b has a solutionhy A must be invertible. [Hint: Is= [²2]Cdrem 4 for A =then the equation Ax = 0 hasny does this imply that A is notsider a = b = 0. Then, if á and= [-]₁ane formula for A works.the vector X =cases of the facts about elemen-lowing Example 5. Here A is a3 x 3 matrix and I = 13. (A general proof would require slightlymore notation.)27. a. Use equation (1)row, (A) = row, (I)29.b. Show that if rows 1 and 2 of A are interchanged, then theresult may be written as EA, where E is an elementarymatrix formed by interchanging rows 1 and 2 of 1.28. Show that if row 3 of A is replaced by row3 (A) - 4-row₁(4).the result is EA, where E is formed from / by replacingrow3 (7) by row3 (1)-4 row, (I).31.c. Show that if row 3 of A is multiplied by 5, then the resultmay be written as EA, where E is formed by multiplyingrow 3 of 1 by 5.Find the inverses of the matrices in Exercises 29-32, if they exist.Use the algorithm introduced in this section.[43]1-31110-21 42 -3 4A =011 Icorrect.0]32.32.122-233. Use the algorithm from this section to find the inverses of0 0 000135. Let A =from Section 2.1 to show thatA, for i = 1.2, 3.022♦and37. Let A =2 336. [M] Let A =003athird columns of A231 530.Let A be the corresponding n x n matrix, and let B be itsinverse. Guess the form of B, and then prove that AB = Iand BA = 1.134. Repeat the strategy of Exercise 33 to guess the inverse of1011I..11['$420115 10¹9]700111 -2T36 -4-2 -7 -92 5 6 . Find the third column of A1 3 4without computing the other columns.09]4 -701I-25 -9 -27546 180537 . Find the second and154 50 149without computing the first column.Construct a 2 x 3 matrix C (by trial anderror) using only 1,-1, and 0 as entries, such that CA = 1₂.Compute AC and note that AC # 13.38. Let A = [II1Prove that your guess isConstruct a 4 x 2 matrix Dusing only 1 and 0 as entriesble that CA = 1₁ for some =.005002,002 .004.001 .002with flexibility measuredthat forces of 30, 50, ar2, and 3. respectively, incorresponding deflections39. Let D =40. [M] Compute the stiffnesList the forces needed topoint 3, with zero deflec41. [M] Let D =.0040.00300010.00052.3 CHAR

Question

33,37 ix Algebra~1/ X], then X = A¹B... then row reduction of [A B] is muchboth A¹ and AB.here B and Caren xp matrices and AB = C. Is this true, in general, when0, where B and Care in x # matricesw that B = C.invertible x n matrices. Show thatby producing a matrix D such thatABC) = 1.xn, B is invertible, and AB is invert-rtible. [Hinr: Let C = AB, and solve= BC for A, assuming that A, B, andvertible.and A = PBP. Solve for B innvertible matrices, does the equationhave a solution, X? If so, find it.rex matrices with A, X, andsupposeertible.u need to invert a matrix, explainertible.of an n x n matrix A are linearlyvertible.(3)anx matrix A span R&#34; wheniew Theorem 4 in Section 1.4.]he equation Ax = 0 has only theby A has n pivot columns and A isheorem 7, this shows that A muste and Exercise 24 will be cited ine equation Ax=b has a solutionhy A must be invertible. [Hint: Is= [²2]Cdrem 4 for A =then the equation Ax = 0 hasny does this imply that A is notsider a = b = 0. Then, if á and= [-]₁ane formula for A works.the vector X =cases of the facts about elemen-lowing Example 5. Here A is a3 x 3 matrix and I = 13. (A general proof would require slightlymore notation.)27. a. Use equation (1)row, (A) = row, (I)29.b. Show that if rows 1 and 2 of A are interchanged, then theresult may be written as EA, where E is an elementarymatrix formed by interchanging rows 1 and 2 of 1.28. Show that if row 3 of A is replaced by row3 (A) - 4-row₁(4).the result is EA, where E is formed from / by replacingrow3 (7) by row3 (1)-4 row, (I).31.c. Show that if row 3 of A is multiplied by 5, then the resultmay be written as EA, where E is formed by multiplyingrow 3 of 1 by 5.Find the inverses of the matrices in Exercises 29-32, if they exist.Use the algorithm introduced in this section.[43]1-31110-21 42 -3 4A =011 Icorrect.0]32.32.122-233. Use the algorithm from this section to find the inverses of0 0 000135. Let A =from Section 2.1 to show thatA, for i = 1.2, 3.022♦and37. Let A =2 336. [M] Let A =003athird columns of A231 530.Let A be the corresponding n x n matrix, and let B be itsinverse. Guess the form of B, and then prove that AB = Iand BA = 1.134. Repeat the strategy of Exercise 33 to guess the inverse of1011I..11['$420115 10¹9]700111 -2T36 -4-2 -7 -92 5 6 . Find the third column of A1 3 4without computing the other columns.09]4 -701I-25 -9 -27546 180537 . Find the second and154 50 149without computing the first column.Construct a 2 x 3 matrix C (by trial anderror) using only 1,-1, and 0 as entries, such that CA = 1₂.Compute AC and note that AC # 13.38. Let A = [II1Prove that your guess isConstruct a 4 x 2 matrix Dusing only 1 and 0 as entriesble that CA = 1₁ for some =.005002,002 .004.001 .002with flexibility measuredthat forces of 30, 50, ar2, and 3. respectively, incorresponding deflections39. Let D =40. [M] Compute the stiffnesList the forces needed topoint 3, with zero deflec41. [M] Let D =.0040.00300010.00052.3 CHAR

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