# Matrix Calculations and Matrix Calculators

#### Term Matrix

The term matrix was introduced in 1850 by James Joseph Sylvester . James Joseph Sylvester (born September 3, 1814 in London, † March 15, 1897) was a British mathematician. Before the matrices already determinants were analyzed as a property of linear systems in the late 16th century by Cardano . Approx. 100 years later Leibniz Her investigations continue on a general basis.

early 200 BC. Chinese mathematicians were the solution methods for systems of equations known 3x3. They recognized that the solution of linear systems depended only on the coefficients. They led to a matrix notation.

as algebraic properties of matrices Carl Friedrich Gauss have been introduced, but not to describe for linear systems, but to linear maps.

The first to systematically investigated matrices as algebraic objects, was Arthur Cayley (1821-1895). He recognized the connection between matrices and systems of linear equations and defined sum, scalar multiple and product for matrices. He continued, the inverse of a matrix.

A complete classification of complex matrices based on the Jordan normal form, which was introduced by Camille Jordan (1838-1922).

As matrix is called a system of elements a ij , which are arranged in a 2-dimensional rectangular pattern. If the scheme of m-rows and n-columns then one speaks of a (m, n)-matrix. The position of an element within the matrix is characterized by two subscripts. The first index is the row number and the second index is the column number. The numbering starts at the top left of the matrix and going from left to right and from top to bottom.

Applies to a matrix n = m it is referred to as the matrix square matrix .

#### Main Diagonal

The elements of the matrix for the subscripts, i = j, ie, the elements a ii are the diagonal elements. While the entire the main diagonal of the matrix. The elements from the lower left to upper right are referred to as secondary diagonal. When speaking of the main diagonal n so you closes the series with a parallel to main diagonal. This also applies for the secondary diagonals.

#### Unit Matrix

The matrix in which all elements of the main diagonal equal to 1 and all other elements are equal to 0 means unit matrix E .

#### Transposed Matrix

The mirrored on the diagonal matrix is called a matrix transpose. For a matrix A = (a ij ) is given by the transposed matrix A T = (a ji ). The transposed of a transposed matrix, the matrix itself is that A = (AT)T.

#### Determinant

Each square matrix can be assigned a unique number, which is referred to as the determinant (det (A)) of the matrix.

In general, the determinant is calculated by multiplying the elements of each diagonal, and then add the values​​. Them the values ​​of the secondary diagonals are subtracted.

Major determinants of one leads to the Laplace development set back to a lower determinants. In this method according to the determinant of any row or column is developed.

#### Calculation Rules for Matrices

The matrix multiplication is associative:

A *( B * C ) = ( A * B ) * C

The matrix multiplication and matrix addition are distributive:

A *( B + C ) = A * B + A * C

For addition and multiplication by real numbers λ, μ:

(λ+ μ)A = λA + μB

and: λ (A + B) = λA + λB

There are zero divisor Matrices A ≠ 0 und B ≠ 0 applies to

A * B = 0

Applies to square matrices

det(A * B) = det(A) * det(B)

#### Inverse Matrix

The inverse matrix A -1 is defined by the following equation

A * A-1 = E

Matrices for which an inverse exists is referred to as regular matrices. Matrices which have no inverse are called singular matrices.

For the inverse matrix, the following calculation rules are valid:

(A * B)-1 = A-1 * B-1

((A)-1)-1 = A

The calculation of the inverse matrix A -1 is either by means of the Gauss-Jordan algorithm or on the adjuncts. The Gauss-Jordan method transforms the matrix (A | E) in the form (E | A -1 ) from the one A -1 can be read directly. With the adjuncts and the determinant of the inverse can be specified directly as A-1=1/det(A) * adj(A)T.

#### Clases of Matrices

A square matrix A is called symmetric matrix if and only if A T = A antisymmetric matrix applies if A T = A orthogonal matrix if and only if AT = A-1

The adjunct of matrix A is calculated in a way that for each matrix element aij is set a sub determinant with removing the line i and the column j. The value of this determinat is multiplied by (-1)i+j that gives the element i,j of the adjungate matrix.

## Matrix Calculations

### Matrix Summation

The addition of two matrices A and B is done by adding the elements of the matrices. C = A + B with c i, j = a i, j + b i, j

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)+$

$\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \dots & {b}_{1m}\\ {b}_{21}& {b}_{22}& \dots & {b}_{2m}\\ & ⋮\\ {b}_{n1}& {b}_{n2}& \dots & {b}_{nm}\end{array}\right)=$

$\left(\begin{array}{cccc}{a}_{11}+{b}_{11}& {a}_{12}+{b}_{12}& \dots & {a}_{1m}+{b}_{1m}\\ {a}_{21}+{b}_{21}& {a}_{22}+{b}_{22}& \dots & {a}_{2m}+{b}_{2m}\\ & ⋮\\ {a}_{n1}+{b}_{n1}& {a}_{n2}+{b}_{n2}& \dots & {a}_{nm}+{b}_{nm}\end{array}\right)$

Calculator for the addition of two matrices.

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### Matrices Subtraction

The subtraction of two matrices A and B is by subtracting the elements of the matrices. C = A - B with c i, j = a i, j - b i, j

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)-$

$\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \dots & {b}_{1m}\\ {b}_{21}& {b}_{22}& \dots & {b}_{2m}\\ & ⋮\\ {b}_{n1}& {b}_{n2}& \dots & {b}_{nm}\end{array}\right)=$

$\left(\begin{array}{cccc}{a}_{11}-{b}_{11}& {a}_{12}-{b}_{12}& \dots & {a}_{1m}-{b}_{1m}\\ {a}_{21}-{b}_{21}& {a}_{22}-{b}_{22}& \dots & {a}_{2m}-{b}_{2m}\\ & ⋮\\ {a}_{n1}-{b}_{n1}& {a}_{n2}-{b}_{n2}& \dots & {a}_{nm}-{b}_{nm}\end{array}\right)$

Calculator for subtraction of two matrices.

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### Matrix multiplication by a scalar

Multiplying a matrix by a scalar is by multiplying each by the scalar matrix elements. a * B = a * bi,j

$\lambda \cdot \left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)=\left(\begin{array}{cccc}\lambda \cdot {a}_{11}& \lambda \cdot {a}_{12}& \dots & \lambda \cdot {a}_{1m}\\ \lambda \cdot {a}_{21}& \lambda \cdot {a}_{22}& \dots & \lambda \cdot {a}_{2m}\\ & ⋮\\ \lambda \cdot {a}_{n1}& \lambda \cdot {a}_{n2}& \dots & \lambda \cdot {a}_{nm}\end{array}\right)$

Calculate the matrix multiplication by a scalar.

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### Matrix Multiplication

The multiplication of two matrices A and B requires that the number of columns of the first matrix is equal to the number of rows of the second matrix. The product obtained by multiplying the row and column elements and summed. For the first element of the result matrix, the elements of the first row of the first matrix are multiplied by the elements of the first column of the second matrix and summed. For the other elements the same for the other rows and columns.

$\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)\cdot$

$\left(\begin{array}{cccc}{b}_{11}& {b}_{12}& \dots & {b}_{1j}\\ {b}_{21}& {b}_{22}& \dots & {b}_{2j}\\ & ⋮\\ {b}_{m1}& {b}_{m2}& \dots & {b}_{mj}\end{array}\right)=$

$\left(\begin{array}{cccc}\sum _{k=1}^{m}\left({a}_{1k}\cdot {b}_{k1}\right)& \sum _{k=1}^{m}\left({a}_{1k}\cdot {b}_{k2}\right)& \dots & \sum _{k=1}^{m}\left({a}_{1k}\cdot {b}_{kj}\right)\\ \sum _{k=1}^{m}\left({a}_{2k}\cdot {b}_{k1}\right)& \sum _{k=1}^{m}\left({a}_{2k}\cdot {b}_{k2}\right)& \dots & \sum _{k=1}^{m}\left({a}_{2k}\cdot {b}_{kj}\right)\\ & ⋮\\ \sum _{k=1}^{m}\left({a}_{nk}\cdot {b}_{k1}\right)& \sum _{k=1}^{m}\left({a}_{nk}\cdot {b}_{k2}\right)& \dots & \sum _{k=1}^{m}\left({a}_{nk}\cdot {b}_{kj}\right)\end{array}\right)$

Calculator for the multiplication of two square 3x3 matrices.

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Calculator for the multiplication of a 2x4 matrix with a 4x2 matrix.

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### Determinant of a Matrix

The determinant of a square matrix is computed by subtracting the sum of the products of the main diagonal of the sum of the products of the secondary diagonal.

Rechner für die Determinante einer 3x3 Matrix.

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### Calculation of the Inverse by Gauss-Jordan

Wanted is the inverse matrix A -1 to the matrix A. For this, first with the identity matrix, the matrix E (A | E) is formed. By suitable transformations we managed to form the (E | A -1 ). In the following the steps of an example can be performed.

$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1N}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2N}\\ & ⋮\\ {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right)$

Gauss-Jordan approach

$\left(A\right|E)=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1N}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2N}\\ & ⋮\\ {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right|\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ & ⋮\\ 0& 0& \dots & 1\end{array})$

Transformations to get the following shape.

$\left(E\right|{A}^{-1})=$ $\left(\begin{array}{cccc}1& 0& \dots & 0\\ 0& 1& \dots & 0\\ & ⋮\\ 0& 0& \dots & 1\end{array}\right|\begin{array}{cccc}{b}_{11}& {b}_{12}& \dots & {b}_{1N}\\ {b}_{21}& {b}_{22}& \dots & {b}_{2N}\\ & ⋮\\ {b}_{N1}& {b}_{N2}& \dots & {b}_{NN}\end{array})$

### Calculator for the inverse matrix

#### Note

The computer does not verify the invertibility or the conditioning of the matrix. A valid result is when the last computation step is the identity matrix is on the left. Otherwise, can possibly be produced by interchanging rows or columns solvability.

Dimension of the matrix N =

Number of digits =

### Calculation of the adjungate matrix

The adjunct of matrix A is calculated in a way that for each matrix element aij is set a sub determinant with removing the line i and the column j. The value of this determinat is multiplied by (-1)i+j that gives the element i,j of the adjungate matrix.

${a}_{ij}^{*}={\left(-1\right)}^{\left(i+j\right)}\left|\begin{array}{ccccccc}{a}_{11}& {a}_{12}& \dots & {a}_{1,j-1}& {a}_{1,j+1}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{i-1,1}& {a}_{i-1,2}& \dots & {a}_{i-1,j-1}& {a}_{i-1,j+1}& \dots & {a}_{i-1,n}\\ {a}_{i+1,1}& {a}_{i+1,2}& \dots & {a}_{i+1,j-1}& {a}_{i+1,j+1}& \dots & {a}_{i+1,n}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{n,j-1}& {a}_{n,j+1}& \dots & {a}_{nn}\end{array}\right|$

The result is the adjungate matrix.

$\mathrm{adj\left(}A\right)=\left(\begin{array}{cccc}{a}_{11}^{*}& {a}_{12}^{*}& \dots & {a}_{1n}^{*}\\ {a}_{21}^{*}& {a}_{22}^{*}& \dots & {a}_{2n}^{*}\\ & ⋮\\ {a}_{n1}^{*}& {a}_{n1}^{*}& \dots & {a}_{nn}^{*}\end{array}\right)$

### Calculator for calculation of the adjungate matrix

$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1N}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2N}\\ & ⋮\\ {a}_{N1}& {a}_{N2}& \dots & {a}_{NN}\end{array}\right)$

Dimension of the matrix N =

Number of digits =