Can a non homogeneous system have a unique solution?

For a non-homogeneous system either (1) the system has a single (unique) solution; (2) the system has more than one solution; (3) the system has no solution at all. We can write these coefficients in the form of a matrix, the coefficient matrix of the linear system.

What is the relationship between the rank of a matrix and the solution of the corresponding system of equations?

If the rank is m then the vectors are linearly independent. If the rank is less than m, then the vectors are linearly dependant. Given the linear system Ax = B and the augmented matrix (A|B). If rank(A) = rank(A|B) = the number of rows in x, then the system has a unique solution.

What is the relationship between homogeneous and solution?

A homogeneous mixture has the same uniform appearance and composition throughout. Many homogeneous mixtures are commonly referred to as solutions. A heterogeneous mixture consists of visibly different substances or phases. The three phases or states of matter are gas, liquid, and solid.

What is a non unique solution?

Non-uniqueness of solutions produces the problem of finding one of them larger than any other between the lower and upper solutions, α and β. Similarly, it is reasonable to look for a solution smaller than any other one. Such solutions are called maximal and minimal solutions.

Is rank of a matrix unique?

Before we can talk about matrix rank, we have to talk about row rank, which is the dimension of row space of the matrix. You should have proven that row rank is unique. Similarly, column rank is unique. From the result that row rank is equal to column rank, we can then talk about matrix rank.

What is the difference between homogeneous and heterogeneous solution?

By definition, a pure substance or a homogeneous mixture consists of a single phase. A heterogeneous mixture consists of two or more phases. When oil and water are combined, they do not mix evenly, but instead form two separate layers.

How are homogeneous solution different from heterogeneous solution?

A homogenous mixture is that mixture in which the components mix with each other and its composition is uniform throughout the solution. A heterogenous mixture is that mixture in which the composition is not uniform throughout and different components are observed.

How do you know if a system has a unique solution?

A nxn homogeneous system of linear equations has a unique solution if and only if its Determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. A system has a unique solution when it is consistent and the number of variables is equal to the number of nonzero rows.

Can a non square matrix have a unique solution?

If its rank is also equal to the number of rows, then you have one unique solution.