The row-echelon form of a matrix is highly useful for many applications. For example, it can be used to geometrically interpret different vectors, solve systems of linear equations, and find out properties such as the determinant of the matrix.

A matrix is a very useful way of representing numbers in a block format, which you can then use to solve a system of linear equations. If you only have two variables, you will probably use a different method. See Solve a System of Two Linear Equations and Solve Systems of Equations for examples of these other methods. But when you have three or more variables, a matrix is ideal. By

The null space of a matrix A{\displaystyle A} is the set of vectors that satisfy the homogeneous equation Ax=0.{\displaystyle A\mathbf {x} =0.} Unlike the column space Col⁡A,{\displaystyle \operatorname {Col} A,} it is not immediately obvious what the relationship is between the columns of A{\displaystyle A} and Nul⁡A.{\displaystyle \operatorname {Nul} A.} Every matrix has a trivial

The matrix equation Ax=b{\displaystyle A\mathbf {x} =\mathbf {b} } involves a matrix acting on a vector to produce another vector. In general, the way A{\displaystyle A} acts on x{\displaystyle \mathbf {x} } is complicated, but there are certain cases where the action maps to the same vector, multiplied by a scalar factor. Eigenvalues and eigenvectors have immense applications in the