Linear system of equations
Regarding the introduction of matrices, we can find their connection with the linear system of $m$ equations with $n$ unknowns:
$$a_{11}\cdot x_1 + a_{12} \cdot x_2 + a_{13} \cdot x_3 + \ldots + a_{1n} \cdot x_n = b_1$$
$$a_{21}\cdot x_1 + a_{22} \cdot x_2 + a_{23} \cdot x_3 + \ldots + a_{2n} \cdot x_n = b_2$$
$$a_{31}\cdot x_1 + a_{32} \cdot x_2 + a_{33} \cdot x_3 + \ldots + a_{3n} \cdot x_n = b_3$$
$$\vdots $$
$$a_{m1}\cdot x_1 + a_{m2} \cdot x_2 + a_{m3} \cdot x_3 + \ldots + a_{mn} \cdot x_n = b_m,$$
where $a_{ij}$ are coefficients of the system and $b_1, \ldots , b_m$ are free members.
The system of equations above we usually connect with the following matrices:
$$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \ldots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} \end{bmatrix} , \quad \mathbf{X}= \begin{bmatrix} x_1 \\ x_2\\ x_3\\ \vdots \\ x_n \end{bmatrix}, \quad \mathbf{B}= \begin{bmatrix} b_1 \\ b_2\\ b_3\\ \vdots \\ b_m \end{bmatrix},$$
where the matrix $\mathbf{A}$ is called a matrix of the system, the matrix $\mathbf{X}$ is a matrix of unknown values and the matrix $\mathbf{B}$ is a matrix of free members.
An equivalent record for the system above is:
$$\mathbf{A} \mathbf{X} = \mathbf{B}.$$
The solution of the matrix system above is a column matrix
$$\mathbf{C} = \begin{bmatrix} \gamma_1 \\ \gamma_2 \\ \gamma_3 \\ \vdots \\ \gamma_{n} \end{bmatrix},$$
for which the substitution $\mathbf{X} = \mathbf{C}$ gives the equality
$$\mathbf{A} \mathbf{C} = \mathbf{B}.$$
The question of existence of the inverse of the matrix often occurs when solving systems of linear equations. If the matrix $\mathbf{A}$ is a regular matrix then by multiplying the equation $\mathbf{A}\mathbf{X}=\mathbf{B}$ to the left with $\mathbf{A^{-1}}$ we obtain the following equation:
$$\mathbf{X} = \mathbf{A^{-1}} \mathbf{B},$$
which gives a unique solution of the system.
Therefore, if we know the inverse matrix $\mathbf{A^{-1}}$ of a matrix $\mathbf{A}$, then the solution of the system of $n$ equations and $n$ unknown values we can write in the form $\mathbf{X} = \mathbf{A^{-1}} \mathbf{B}$, which makes sense only if the matrix of the system is invertible.
An augmented matrix $\mathbf{\hat{A}}$ for the matrix system $\mathbf{A} \mathbf{X} = \mathbf{B}$ is defined as
$$\mathbf{\hat{A}} = [\mathbf{A} | \mathbf{B}] =$$
$$\left[ \begin{array} {ccccc|c} a_{11} & a_{12} & a_{13} & \ldots & a_{1n} & b_{1}\\ a_{21} & a_{22} & a_{23} & \ldots & a_{2n} & b_{2}\\ a_{31} & a_{32} & a_{33} & \ldots & a_{3n} & b_{3}\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & a_{m3} & \ldots & a_{mn} & b_{m}\end{array} \right].$$
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 $0$,
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
$$\mathbf{A} =\begin{bmatrix} 3 & 2 & 7 \\ 0 & 5 & 2 \\ 0 & 0 & 6 \end{bmatrix}$$
is in an echelon form, and a matrix
$$\mathbf{B} =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
is in reduced echelon form.
For the purposes of finding the inverse matrix of a square matrix $\mathbf{A}$ of order $n$ by using the method of elementary transformations, we will observe the matrix $\mathbf{[A|I]}$, where the matrix $\mathbf{I}$ is an identity matrix, also of order $n$.
Rank of matrices
The rank of matrices is another concept appropriate for recognizing the regularity of matrices and calculating their inverses.
Let $\mathbf{A} = [a_{ij}] \in \mathbb{R}^{m \times n}$ and let $\mathbf{C_{1}} = \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \\ \vdots \\ a_{m1} \end{bmatrix}$ , $\mathbf{C_{2}} =\begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \\ \vdots \\ a_{m2} \end{bmatrix}$ , $\ldots$ , $\mathbf{C_{n}} =\begin{bmatrix} a_{1n} \\ a_{2n} \\ a_{3n} \\ \vdots \\ a_{mn} \end{bmatrix}$
be columns of the matrix $\mathbf{A}$. The rank of the matrix $\mathbf{A}$, $r (\mathbf{A})$, is defined by the formula:
$r(\mathbf{A}) = dim [ \mathbf{C_{1}} , \mathbf{C_{2}}, \ldots, \mathbf{C_{n}}].$
This means that the rank is the number of linearly independent columns in the row echelon form of the matrix $\mathbf{A}$, that is, the number of the non-zero rows, obtained by using elementary operations on the matrix $\mathbf{A}$.
A zero matrix is the only matrix with the rank $0$. An identity matrix has a full rank, that is $r(\mathbf{I}) = n$.
A matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ and its transpose matrix $\mathbf{A}^T$ have the same rank, that is, $r(\mathbf{A}) = r(\mathbf{A}^T)$.
Let the matrix $\mathbf{A’}$ be a matrix obtained from the matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ by applying one of the elementary transformations. Then is $r(\mathbf{A’}) = r(\mathbf{A})$.
For instance, the rank of the matrix $\mathbf{A} =\begin{bmatrix} 3 & 2 & 7 \\ 0 & 5 & 2 \\ 0 & 0 & 6 \end{bmatrix}$ is 3, and the rank of matrix $\mathbf{B} =\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ is 2.
Solutions of linear systems
Linear systems of equations have either no solution, one solution or infinitely many solutions.
The Kronecker – Capelli theorem:
The system $\mathbf{A}\mathbf{X} = \mathbf{B}$ is solvable iff $r (\mathbf{A}) = r (\mathbf{\hat{A}})$.
The system $\mathbf{A}\mathbf{X} = \mathbf{B}$ is homogeneous if $b_{1} = b_{2} = \ldots = b_{m} = 0$, that is
$$\mathbf{A}\mathbf{X} = 0.$$
A homogeneous system is always solvable. If $r(\mathbf{A})= n$ then the system $\mathbf{A}\mathbf{X} = 0$ has only a trivial solution.
Cramer’s rule:
If the matrix $\mathbf{A}$ of the system $\mathbf{A} \mathbf{X} = \mathbf{B}$ is invertible, then the solution of the system is unique and given with the formula:
$$x_{i}=\frac{D_i}{D}, \quad i = 1, 2, \ldots, n ,$$
where $D = det \mathbf{A}$ and $D_{i}$ is the determinant of the matrix obtained by the replacing $i$-th column of matrix $\mathbf{A}$ by the column matrix $\mathbf{B}$.
Let us consider one more special case. We observe the system $\mathbf{A}\mathbf{X} = \mathbf{B}$ in which the number of equations $m$ is equal to the number of unknown values $n$, that is, matrix $\mathbf{A}$ is a square matrix. If $r(\mathbf{A}) < n$ then is $n- r(\mathbf{A}) > 0$ and we have an infinite set of solutions.
