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 the following matrices:
where the matrix is called a matrix of the system, the matrix is a matrix of unknown values and the matrix is a matrix of free members.
An equivalent record for the system above is:
The solution of the 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 the 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 the matrix , where the 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 the matrix . The rank of the matrix , , is defined by the formula:
This means that the rank is the number of linearly independent columns in the row echelon form of the matrix , that is, the number of the non-zero rows, obtained by using elementary operations on the 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 the matrix be a matrix obtained from the matrix by applying one of the elementary transformations. Then is .
For instance, the rank of the 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 the system has only a trivial solution.
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 the column matrix .
Consider one more special case. We observe the 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.