Is it possible for Ax B to have no solution?

Is it possible for Ax B to have no solution?

1. If A is a square matrix, then if A is invertible every equation Ax = b has one and only one solution. Namely, x = A’b. If A is not invertible, then Ax = b will have either no solution, or an infinite number of solutions.

Does Ax B have a unique solution?

The system AX = B has a unique solution provided dim(N(A)) = 0. Since, by the rank theorem, rank(A) + dim(N(A)) = n (recall that n is the number of columns of A), the system AX = B has a unique solution if and only if rank(A) = n. A linear system of the form AX = 0 is said to be homogeneous.

What does no unique solution mean?

A system of linear equations can have no solution, a unique solution or infinitely many solutions. A system has no solution if the equations are inconsistent, they are contradictory. for example 2x+3y=10, 2x+3y=12 has no solution. is the rref form of the matrix for this system.

What does it mean if Ax B has a unique solution?

Theorem 1. Let A be a square n × n matrix. Then Ax = b has a unique solution if and only if the only solution of Ax = 0 is x = 0. A rephrasing of this is (in the square case) Ax = b has a unique solution exactly when {A1,A2,…,An} is a linearly independent set.

How do you know if Ax B has infinite solutions?

Ax = b has a unique solution if and only if rank[A] = rank[A|b] = n. iii. Ax = b has infinitely many solutions if and only if rank[A] = rank[A|b] < n. To illustrate this theorem, let’s look at the simple systems below.

How do you find solutions to Ax B?

One way to find a particular solution to the equation Ax = b is to set all free variables to zero, then solve for the pivot variables. The general solution to Ax = b is given by xcomplete = xp + xn, where xn is a generic vector in the nullspace.

How do you know if Ax B has a solution?

Ax = b has a solution if and only if b is a linear combination of the columns of A. Theorem 4 is very important, it tells us that the following statements are either all true or all false, for any m × n matrix A: (a) For every b, the equation Ax = b has a solution.

What is the solution of the equation Ax B 0?

b = 0 – identity equation, the solution is any x number belonging to the set of real numbers (infinitely many solutions), b ≠ 0 – b ≠ 0 – contradictory equation (no solutions).

How do you know if an equation has a unique solution?

In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.

Can non square matrices have unique solutions?

In general you can have zero, one or an infinite number of solutions to a linear system of equations, depending on its rank and nullity relationship.

What does it mean when Ax B has infinitely many solutions?

The equation Ax = b has an infinite number of solutions. The equation Ax = 0 has a non-zero solution. Null(A) = {0}. A non–zero linear combination of the columns of A is equal to 0.

When does ax b have a unique solution?

Ax = b has a solution if and only if b is a linear combination of the columns of A. Note: If A does not have a pivot in every row, that does not mean that Ax = b does not have a solution for some given vector b. It just means that there are some vectors b for which Ax = b does not have a solution.

Are there any free variables in solving Ax = b?

There are no free variables or special solutions. If Ax = b has a solution, it is unique; there is either 0 or 1 solution. Exam­ ples like this, in which the columns are independent, are common in applica­ tions. We know r ≤ m, so if r = n the number of columns of the matrix is less than or equal to the number of rows.

Are there any vectors for which Ax does not have a solution?

It just means that there are some vectors b for which Ax = b does not have a solution. ˆ Finally, it is very useful to know that multiplying a vector by a vector has the following nice properties: (a) A(u+ v) = A(u) + A(v), for vectors u;v (b) A(cu) = cA(u), for vectors u and scalars c.

When does a system have a unique solution?

A system has a unique solution when it is consistent and the number of variables is equal to the number of nonzero rows. What is the row vector rule? Row-vector rule for computing Ax If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.