13 +If A is larger than 2 x 2, then row reduction of [A B] is muchfaster than computing both A and AB.13. Suppose AB = AC, where B and C are n x p matrices and Ais invertible. Show that B = C. Is this true, in general, whenA is not invertible?14. Suppose (BC)D = 0, where B and C are m x n matricesand D is invertible. Show that B = C.2015. Suppose A, B, and C are invertible n x n matrices. Show thatABC is also invertible by producing a matrix D such that(ABC) D = I and D (ABC) = I.JOU16. Suppose A and B are n xn, B is invertible, and AB is invert-ible. Show that A is invertible. [Hint: Let C = AB, and solveORTOVOLthis equation for A.]to nemote117. Solve the equation AB = BC for A, assuming that A, B, andC are square and B is invertible.18. Suppose P is invertible and A = PBP-¹. Solve for B interms of A.en2019. If A, B, and Care n x n invertible matrices, does the equationC-¹(A + X)B¯¹ = In have a solution, X? If so, find it.20. Suppose A, B, and X are n x n matrices with A, X, andA - AX invertible, and suppose(A - AX)¯¹ = X-¹ B(3)a. Explain why B is invertible.to b. Solve (3) for X. If you need to invert a matrix, explainwhy that matrix is invertible.21. Explain why the columns of an n x n matrix A are linearlyindependent when A is invertible.22. Explain why the columns of an n x n matrix A span R" whenA is invertible. [Hint: Review Theorem 4 in Section 1.4.]23. Suppose A is n x n and the equation Ax = 0 has only thetrivial solution. Explain why A has n pivot columns and A isrow equivalent to In. By Theorem 7, this shows that A mustSection 2.3.)be invertible. (This exercise and Exercise 24 will be cited in24. Suppose A is n x n and the equation Ax = b has a solutionA row equivalent to I,,?]for each b in R". Explain why A must be invertible. [Hint: Is27. a. Use equarow, (A) =b. Show thatresult mamatrix for28. Show that if rthe result isrow3 (I) by r29.c. Show thamay be wrow 3 of,Find the inversesUse the algorithm2sill lo31.1 0-3 1L2-333. Use the algor1011Let A be theinverse. Gueand BA = I34. Repeat the s1A = 1correct.35. Let A =without comp

Question

13 +If A is larger than 2 x 2, then row reduction of [A B] is muchfaster than computing both A and AB.13. Suppose AB = AC, where B and C are n x p matrices and Ais invertible. Show that B = C. Is this true, in general, whenA is not invertible?14. Suppose (BC)D = 0, where B and C are m x n matricesand D is invertible. Show that B = C.2015. Suppose A, B, and C are invertible n x n matrices. Show thatABC is also invertible by producing a matrix D such that(ABC) D = I and D (ABC) = I.JOU16. Suppose A and B are n xn, B is invertible, and AB is invert-ible. Show that A is invertible. [Hint: Let C = AB, and solveORTOVOLthis equation for A.]to nemote117. Solve the equation AB = BC for A, assuming that A, B, andC are square and B is invertible.18. Suppose P is invertible and A = PBP-¹. Solve for B interms of A.en2019. If A, B, and Care n x n invertible matrices, does the equationC-¹(A + X)B¯¹ = In have a solution, X? If so, find it.20. Suppose A, B, and X are n x n matrices with A, X, andA - AX invertible, and suppose(A - AX)¯¹ = X-¹ B(3)a. Explain why B is invertible.to b. Solve (3) for X. If you need to invert a matrix, explainwhy that matrix is invertible.21. Explain why the columns of an n x n matrix A are linearlyindependent when A is invertible.22. Explain why the columns of an n x n matrix A span R&#34; whenA is invertible. [Hint: Review Theorem 4 in Section 1.4.]23. Suppose A is n x n and the equation Ax = 0 has only thetrivial solution. Explain why A has n pivot columns and A isrow equivalent to In. By Theorem 7, this shows that A mustSection 2.3.)be invertible. (This exercise and Exercise 24 will be cited in24. Suppose A is n x n and the equation Ax = b has a solutionA row equivalent to I,,?]for each b in R&#34;. Explain why A must be invertible. [Hint: Is27. a. Use equarow, (A) =b. Show thatresult mamatrix for28. Show that if rthe result isrow3 (I) by r29.c. Show thamay be wrow 3 of,Find the inversesUse the algorithm2sill lo31.1 0-3 1L2-333. Use the algor1011Let A be theinverse. Gueand BA = I34. Repeat the s1A = 1correct.35. Let A =without comp

Accepted Answer

To prove B=C when A is not invertible.

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