9. We defined an n x n matrix to be invertible if there is a matrix B such that BA = In. In this exercise, we will explain why B is also invertible and that I. This means that, if B = A-1, then A = B-1. АВ Given the fact that BA In, explain why the matrix B must also be a а. square n x n matrix. I, it follows that b. Suppose that b is a vector in R". Since we have BA B(Ab) = b. Use this to explain why the columns of B span R". What does this say about the pivot positions of B? c. Explain why the equation Bx o has only the trivial solution. I, multiply both sides by B to obtain d. Beginning with the equation, BA BAB = B. We will rearrange this equation: ВАВ — В ВАВ — В — 0 B(AB – I) = 0. o has only the trivial solution, Since the homogeneous equation Bx explain why AB – I = 0 and therefore, AB = I. %3D

Q: 11. A matrix ME Mnxn (C) is called skew-symmetric if Mt = -M. Prove that if M is skew-symmetric and…

A: 11. A matrix M∈Mn×nℂ is skew symmetry and n is odd. We have to prove that M is not invertible. Also,…

Q: The mist Consider ann x m matrix A. Find dim(im(A)) + dim(ker(A")), in terms of m and n.

A: Please check step 2 for the solution.!

Q: 5. For each of the statements given below decide if it is true or false. If it is true explain why.…

A:

Q: 9. Assume that matrix A is row equivalent to matrix B, and -1 B = 0 -2 1 -4 A 2 -4 1 -6 5 -6 10 7…

A: Row space of the matrix A is the space of vectors which is generated by the linear combination of…

Q: 2) An n x n matrix A is called nilpotent if Ak = 0 for some positive integer k. Show that if A + 0…

A:

Q: Let A be a matrix whose entries are integers, and det A=1. 4 : Show that the entries of A-1 are all…

A: given ,det A = 1

Q: Show that the length of Ax equals the length of ATx if AAT = ATA. A is a matrix, T is its transpose

A: Show that the length of Ax equals the length of ATx  if AAT = ATA. A is a matrix, T is its transpose

Q: If A is a 31x24 matrix whose rank equals 10, find dim(Nul(AT)). Your Answer:

A:

Q: 3. Let A = 3-1 1 V₁ = 3 -2 2 = 2 V₂ = 3 V3 = 6 V4 and use the facts Av₁ = V₁ AV₂ = V₂ AV3 = V3 V4 €…

A: 3. Let , and use the facts  to answer the following question.(c) If A is diagonalizable then give…

Q: How do we know that the ith row of an invertible matrix B is orthogonal to the jth column of B if i…

A: We have been given a matrix B. To Prove: ith row of B and jth col of B-1 are orthogonal, where B is…

Q: 7. Let A and B two 3-by-3 matrices, with |B| = 1. Let C be the inverse matrix of B. Which of the…

A:

Q: . Can a square matrix with two identical columns be invertible? Prove it clearly.

A:

Q: 4. Diagonalize the matrix, if possible. That is, find an invertible matrix P and a diagonal matrix D…

A:

Q: 2. Let A be the matrix shown below A = [22] :) Find the matrix P that diagonalizes A.

A:

Q: b) Consider the following matrices: Solution: M₁ = [1 −7], M₂ = [1 2], M₁ = (1 1]. A = [23], B= [13…

A: Step 1: Determine if A is in the set Span({M1​,M2​,M3​}).Step 2: Determine if B is in the set…

Q: es m- Exercise 2.3.34 If A and B are n x n matrices, show that: a. AB = BA if and only if (A+B)² =…

A:

Q: suppose an nxn matrix has positive diagonal entries and negative everywhere else. suppose sum of the…

A: Introduction: To prove that the determinant of matrix A is non-zero, we will show that A is…

Q: Prove that there does not exist a 43 × 30 matrix with entries from {0, 1} such that there are 17 is…

A: As per the Guidelines, we can solve only one question. Please repost it for the other part.We can…

Q: Suppose that A is an invertible matrix with n columns, and entry aij in row i and column j. 1.…

A: Let matrix is A and the entries are aij .