Regarding the introduction of matrices, we can find their connection with the linear system of equations with unknowns:

where are coefficients of the system and are free members.

The system of equations above we usually connect with following matrices:

where a matrix is called a matrix of the system, matrix is matrix of unknown values and matrix is a matrix of free members.

An equivalent record for the system above is:

The solution of a matrix system above is a column matrix

for which the substitution gives the equality

The question of existence of the inverse of the matrix often occurs when solving systems of linear equations. If a matrix is a regular matrix then by multiplying the equation to the left with we obtain the following equation:

which gives a unique solution of the system.

Therefore, if we know the inverse matrix of a matrix , then the solution of the system of equations and unknown values we can write in the form , which makes sense only if the matrix of the system is invertible.

**An augmented matrix ** for the matrix system is defined as

An algorithm for solving systems of linear equations is the **Gaussian elimination**. This algorithm can also be used for calculating the determinant of matrices and finding an inverse matrix of a given square matrix.

Gaussian elimination is based on using **elementary operations** to transform the augmented matrix into the upper triangular form.

**The elementary operations** are:

1.) interchanging the position of two rows (or columns),

2.) multiplication of a row (or column) by a scalar different from ,

3.) adding a one row (or column), multiplied by a scalar, to another.

A matrix is in **an echelon form **or** row echelon form **if satisfies the following conditions:

1.) all non – zero rows are located above any rows or all zeroes,

2.) each leading element of a row (**pivot**) is in column located to the right of the leading element of the row above it,

3.) all elements in a column below pivot are zeroes.

A matrix is in **reduced row echelon form** if is an row echelon form and pivot in any non-zero row is 1.

For example, a matrix

is in an echelon form, and a matrix

is in reduced echelon form.

For the purposes of finding the inverse matrix of a square matrix of order by using the method of elementary transformations, we will observe a matrix , where a matrix is an identity matrix also of order .

**Rank of matrices**

The rank of matrices is another concept appropriate for recognizing the regularity of matrices and calculating their inverses.

Let and let , , ,

be columns of a matrix . The rank of matrix , , is defined by the formula:

This means that **the rank** is the number of linearly independent columns in the row echelon form of a matrix , that is, the number of non-zero rows, obtained by using elementary operations on matrix .

A zero matrix is the only matrix with the rank . An identity matrix has a full rank, that is .

A matrix and its transpose matrix have the same rank, that is .

Let a matrix be a matrix obtained from matrix by applying one of the elementary transformations. Then is .

For instance, the rank of matrix is 3, and the rank of matrix is 2.

## Solutions to equations

Linear systems of equations have either no solution, one solution or infinitely many solutions.

**The Kronecker – Capelli theorem**

The system is solvable iff .

The system is **homogeneous** if , that is

A homogeneous system is always solvable. If then a system has only the trivial solution.

**Cramer’s rule**

If the matrix of the system is invertible, then the solution of the system is unique and given with the formula:

where and is the determinant of the matrix obtained by the replacing -th column of matrix by column matrix .

Consider one more special case. We observe system in which the number of equations is equal to the number of unknown values , that is, matrix is a square matrix. If then is and we have an infinite set of solutions.