Latest Tweets

Matrices and systems 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} = \left[ \begin{array} {ccccc}

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{array} \right] , \quad  \mathbf{X}= \left[ \begin{array} {c} x_1 \\

x_2\\

x_3\\

\vdots \\

x_n\\

\end{array} \right] , \quad

\mathbf{B}= \left[ \begin{array} {c} b_1 \\

b_2\\

b_3\\

\vdots \\

b_m\\

\end{array} \right],$$

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} =

\left[\begin{array} {c}

\gamma_1 \\

\gamma_2 \\

\gamma_3 \\

\vdots \\

\gamma_{n} \\

\end{array} \right],$$

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} =\left [ \begin{array}{ccc}

3 & 2 & 7 \\

0 & 5 &  2 \\

0 &  0 & 6 \\

\end{array}\right]$$

is in an echelon form, and a matrix

$$\mathbf{B} =\left [ \begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 &  0 \\

0 &  0 & 0 \\

\end{array}\right]$$

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}} = \left [\begin{array} {c}

a_{11}  \\

a_{21} \\

a_{31} \\

\vdots \\

a_{m1} \\

\end{array}\right]$ , $\mathbf{C_{2}} =\left [\begin{array} {c}

a_{12}  \\

a_{22} \\

a_{32} \\

\vdots \\

a_{m2} \\

\end{array}\right]$ , $\ldots$ , $\mathbf{C_{n}} =\left [\begin{array} {c}

a_{1n}  \\

a_{2n} \\

a_{3n} \\

\vdots \\

a_{mn} \\

\end{array}\right]$

be columns of the matrix $\mathbf{A}$. The rank of the matrix $\mathbf{A}$, $r (\mathbf{A})$, is defined by the formula:

 

$$r(\mathbf{A}) = dim [\left{\mathbf{C_{1}} ,  \mathbf{C_{2}}, \ldots, \mathbf{C_{n}}\right}].$$

 

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} =\left [ \begin{array}{ccc}

3 & 2 & 7 \\

0 & 5 &  2 \\

0 &  0 & 6 \\

\end{array}\right]$ is 3, and the rank of matrix $\mathbf{B} =\left [ \begin{array}{ccc}

1 & 0 & 0 \\

0 & 1 &  0 \\

0 &  0 & 0 \\

\end{array}\right]$ 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 $\mathbf{A}\mathbf{X} = \mathbf{B}$ 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 $\mathbf{B}$.

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.