If A has full rank, then det A is non zero. So
A-1 exists. Thus for non zero x, we get
A-1Ax=0 that is x is a solution, which non zero.
Option c is right.
Exercise 15.3 Under what conditions does the matrix equation Ax = 0 have a solution x...
a. Every matrix equation Ax b corresponds to a vector equation with the same solution set. Choose the correct answer below. O A. False. The matrix equation Ax-b does not correspond to a vector equation with the same solution set. O B. False. The matrix equation Ax b only corresponds to an inconsistent system of vector equations. O c. True. The matrix equation Ax-bis simply another notation for the vector equation x1a1 + x2a2 +·.. + xnan-b, where al ,...
Let A = and b = . Show that the equation Ax = b does not have a solution for some choices of b, and describe the set of all b for which Ax = b does have a solution. How can it be shown that the equation Ax = b does not have a solution for some choices of b? A. Row reduce the augmented matrix [A b] to demonstrate that [A b] has a pivot position in every row B. Find a vector...
1-4 - 31 Let A= 3 and b= Show that the equation Ax=b does not have a solution for all possible b, and describe the set 4 26 of all b for which Ax=b does have a solution. How can it be shown that the equation Ax = b does not have a solution for all possible b? Choose the correct answer below. O A. Row reduce the augmented matrix [ a b ] to demonstrate thatſ A b )...
b. - 2 -1 1 and b Let A = Show that the equation Ax =b does not have a solution for all possible b, and -3 0 3 4-2 2 b3 describe the set of all b for which Ax b does have a solution How can it be shown that the equation Ax = b does not have a solution for all possible b? Choose the correct answer below. O A. Find a vector b for which the...
For problems 4) and 5) answer the following (a) Does the equation Ax = 0 have a nontrivial solution? (b) Does the equation Ax = b have at least one solution for every possible b? 4) A is a 4 x 4 matrix with three pivot positions. 5) A is a 3 x 2 matrix with two pivot positions.
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
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...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
IT a) If one row in an echelon form for an augmented matrix is [o 0 5 o 0 b) A vector bis a linear combination of the columns of a matrix A if and only if the c) The solution set of Ai-b is the set of all vectors of the formu +vh d) The columns of a matrix A are linearly independent if the equation A 0has If A and Bare invertible nxn matrices then A- B-'is the...