Currently workable: Suppose and m x n matrix A has n pivot columns. Prove why, for...

Question

Question

Answer 1

Answer #1

Given that A has n Pivot columns hence

Rank(A) = n consider rank of [A | b] and this a

Can be either equal to rank of A or not.

If Rank (A) = Rank [A|b] = n , number of columns of A then the equation Ax=b has unique solution.

If Rank(A) not equal to Rank [A|b] then the system has no solution. Hence the system Ax = b has atmost one solution

Answer 2

Similar Homework Help Questions

Currently workable: Let A be an n x n invertible matrix. Suppose AB n x p....

Currently workable: Let A be an n x n invertible matrix. Suppose AB n x p. Prove that B = C. AC, where B and Care Is this true in general? If not, state when it is not true and provide a counter- example. + Drag and drop your files or click to browse...
12. a. If there is an n x n matrix D such that AD = 1,...

12. a. If there is an n x n matrix D such that AD = 1, then there is also an n x n matrix C such that CA= 1. b. If the columns of A are linearly independent, then the columns of A span Rn. c. If the equation Ax = b has at least one solution for each bin Rn, then the solution is unique for each b. d. If the linear transformation (x) -> Ax maps Rn into Rn, then...
Suppose A is a square matrix such that det A4 invertible. 0. Prove that A is...

Suppose A is a square matrix such that det A4 invertible. 0. Prove that A is not Suppose that A is a square matrix such that det A" invertible and that it must have determinant 1. 1. Prove that A is Matrices whose determinant is 1 are part of a group (not just the english word, a special math term, ask if you want the deets) called the Special Linear Group, denoted SL(n) + Drag and drop your files or...

Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that...

Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
Suppose a 4x7 coeficient matrix for a system has four pivot columns. Is the system consistent?...

Suppose a 4x7 coeficient matrix for a system has four pivot columns. Is the system consistent? Why or why not? Choose the corect answer below. OA. There is a pivot position in each raw of the coefficient matrix. The auugmented matrix will have five columns and will not have a row of the form so the system is consistent. n o o 0 1 O B. There is a pivot position in each row of the coefficient matrix. The augmented...
True/False: Give a brief justification for your answer a) If an m x n matrix A...

True/False: Give a brief justification for your answer a) If an m x n matrix A has a pivot position in each row, then the equation Ax=b has a unique solution for each b in R^m. b) If {u,v,w} is linearly independent, then u, v, w are not in R^2. c) If A is a 5 x 4 matrix, then the linear transformtion x -> Ax is not onto.

7. Suppose A is a 6 x 3 matrix with 3 pivot positions. (a) Does the...

7. Suppose A is a 6 x 3 matrix with 3 pivot positions. (a) Does the equation Ax O have a nontrivial solution? (b) Does the equation Ax =b have at least one solution for every b E R6? %3D
A.) if A is an m*n matrix, such that Ax=0 for every vector x in R^n, then A is the m * n Zero mat...

a.) if A is an m*n matrix, such that Ax=0 for every vector x in R^n, then A is the m * n Zero matrix b.) The row echelon form of an invertible 3 * 3 matrix is invertible c.) If A is an m*n matrix and the equation Ax=0 has only the trivial solution, then the columns of A are linearly independent. d.) If T is the linear transformation whose standard matrix is an m*n matrix A and the...
Suppose an 8 x 10 matrix A has eight pivot columns. Is Col A=R8? Is Nul...

Suppose an 8 x 10 matrix A has eight pivot columns. Is Col A=R8? Is Nul A=R2? Explain your answers. Is Col A =R8? A. Yes. Since A has eight pivot columns, dim Col A is 8. Thus, Col A is an eight-dimensional subspace of R8, so Col A is equal to R8 OB. No, the column space of Ais not R. Since A has eight pivot columns, dim Col A is 0. Thus, Col A is equal to 0....

Prove the following theorem: If an m x n matrix U has orthonormal columns, then UTU...

Prove the following theorem: If an m x n matrix U has orthonormal columns, then UTU = I.