# Calculatrice matricielle en ligne

## What is Matrix algebra?

• Matrix algebra is a means of making calculations upon arrays of numbers (or data).
• Most data sets are matrix-type

### Why use it?

• Matrix algebra makes mathematical expression and computation easier.
• It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.

## Definitions

1. Matrices – definitions
2. Matrix notation
3. Equal matrices
4. Addition and subtraction of matrices
5. Scalar multiplication of matrices
6. Transpose of a matrix
7. Special matrices
8. Multiplication of two or more matrices
9. Determinant of a square matrix
10. Inverse of a square matrix
11. Solution of a set of linear equations
12. Eigenvalues and eigenvectors

## Matrices – definitions

• A matrixis a set of real or complex numbers (called elements) arranged in rows and columns to form a rectangular array.
• A matrix having m rows and n columns is called an m × n matrix. For example: $\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$ is a 2 × 3 matrix.
• A matrix is a rectangular array of numbers or symbolic elements
In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristics
The dimension of a matrix is its number of rows and columns, often denoted as r x c (r rows by c columns) Can be represented in full form or abbreviated form:
• scalar a scalar is a number (denoted with regular type: 1 or 22).
• Vector A matrix with one column (column vector) or one row (row vector).
• Row vector A row vector consists of a single row. For example: $\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$
• Column vector A column matrix consists of a single column. For example: $\left(\begin{array}{c}6\\ 3\\ 8\end{array}\right)$
• Double suffix notation Each element of a matrix has its own address denoted by double suffices, the first indicating the row and the second indicating the column. For example, the elements of 3 × 4 matrix can be written as:

## Matrix Notation

Where there is no ambiguity a matrix can be represented by a single general element in brackets or by a capital letter in bold type.

## Equal matrices

Two matrices are equal if corresponding elements throughout are equal

Where

## Addition and subtraction of matrices

Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:

## Multiplication of matrices

1. Scalar multiplication
2. Multiplication of two or more matrices

Scalar multiplication

To multiply a matrix by a single number (a scalar), each individual element of the matrix is multiplied by that number. For example;

That is:

Properties of matrix addition and scalar multiplication

Then :-

• $A+BA=ABA+AA$ Commutative law for addition
• $A+\left(B+C\right)=\left(A+B\right)+C$ Associative law for addition
• $\left(cd\right)A=c\left(dA\right)$ Associative law for scalar multiplication
• $1A=A$ Unit element for scalar multiplication
• $c\left(A+B\right)=cA+cB$ Distributive law 1 for scalar multiplication
• $\left(c+d\right)A=cA+dA$ Distributive law 2 for scalar multiplication

Properties of zero matrices

Then

the additive identity for all m×n matrices

the additive inverse of A

## Transpose of a matrix

If a new matrix is formed by interchanging rows and columns the new matrix is called the transpose of the original matrix. For example, if:

Thus

If A = AT, then A is symmetric

## Symmetric matrix

A square matrix A is symmetric if  A = AT

Skew-symmetric matrix:

A square matrix A is skew-symmetric if  AT = A

## TYPES OF MATRICES

 NAME DESCRIPTION EXAMPLE Row matrix A matrix with only 1 row $\left(\begin{array}{cccc}3& 2& 1& 4\end{array}\right)$ Column matrix A matrix with only I column $\left(\begin{array}{c}2\\ 3\end{array}\right)$ Identity matrix Diagonal matrix having each diagonal element equal to one (I) $\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|$ Zero matrix A matrix with all zero entries $\left|\begin{array}{cc}0& 0\\ 0& 0\end{array}\right|$ Upper Triangular matrix Square matrix having all the entries zero below the principal diagonal $\left[\begin{array}{ccc}2& 5& 6\\ 0& 4& 6\\ 0& 0& 7\end{array}\right]$ Lower Triangular matrix Square matrix having all the entries zero above the principal diagonal $\left[\begin{array}{ccc}2& 0& 0\\ 5& 4& 0\\ 3& 6& 7\end{array}\right]$

## Special matrices

• Square matrix
• Diagonal matrix
• Unit matrix (identity matrix)
• Null matrix

Square matrix

• A square matrix is of order m × m.
• A square matrix is symmetric if For example: $\left(\begin{array}{ccc}1& 2& 2\\ 2& 8& 9\\ 5& 9& 4\end{array}\right)$
• A square matrix is skew-symmetric if For example $\left(\begin{array}{ccc}0& 2& 5\\ -2& 0& 9\\ -5& -9& 0\end{array}\right)$

Diagonal Matrices.

A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero

Identity Matrix (Unit Matrix)

• An identity matrix is a diagonal matrix where the diagonal elements all equal one. $I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
• It is named I and it comes in different sizes. $I2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ $I3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Null Matrix (Zero Matrix)

A square matrix where all elements equal zero

$\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

Properties of zero matrices

## Matrix Multiplication

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}5& -2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}5& -2\\ 3& 4\end{array}\right]$
$\left[\begin{array}{cc}5& -2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}5& -2\\ 3& 4\end{array}\right]$

So, the identity matrix multiplied by any matrix lets the “any” matrix keep its identity!

Mathematically, IA = A and AI = A !!

### The Multiplicative Identity

$AI=A$ Multiply $\left[\begin{array}{cc}-2& 5\\ 4& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ $A=\left[\begin{array}{cc}-2& 5\\ 4& 0\end{array}\right]$

### Give the multiplicative identity for matrix B.

This identity matrix is I4.

## Matrix Multiplication constraint

• (m × n) × (p × n) = cannot be done
• (1 × n)  × (n × 1) = a scalar (1x1)

## Trace of a Matrix

Trace: the sum of the diagonal of a square matrix.

### Show that AB and BA are not equal for the matrices

Solve $AB=\left[\begin{array}{cc}1& 3\\ 2& -1\end{array}\right]\left[\begin{array}{cc}2& -1\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}2& 5\\ 4& -4\end{array}\right]$
$BA=\left[\begin{array}{cc}2& -1\\ 0& 2\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}0& 7\\ 4& -2\end{array}\right]$

### An example in which cancellation is not valid

Show that AC=BC C is noninvertible, (i.e., row 1 and row 2 are not independent) Solve $AC=\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}-2& 4\\ -1& 2\end{array}\right]$
$BC=\left[\begin{array}{cc}2& 4\\ 2& 3\end{array}\right]\left[\begin{array}{cc}1& -2\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}-2& 4\\ -1& 2\end{array}\right]$

So  AC=BC
But A≠ B

## Matrix Multiplication Properties

• AB does not necessarily equal BA
• (BA may even be an impossible operation)

For example

• A × B = C (2 × 3) × (3 × 2) = (2 × 2)

• B × A = D (3 × 2) × (2 × 3) = (3 × 3)

• Matrix multiplication is Associative
• Multiplication and transposition

## Matrix Inverse

• Matrix Inverse: Needed to perform the “division” of 2 square matrices
• In scalar terms A/B is the same as A * 1/B
• When we want to divide matrix A by matrix B we simply multiply by A by the inverse of B
• An inverse matrix is defined as