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.
Two matrices are equal if corresponding elements throughout are equal
Where
Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:
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;
Properties of matrix addition and scalar multiplication
Then :-
Then
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
A square matrix A is symmetric if A = AT
Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A
NAME | DESCRIPTION | EXAMPLE |
Row matrix |
A matrix with only 1 row |
|
Column matrix |
A matrix with only I column |
|
Identity matrix |
Diagonal matrix having each diagonal element equal to one (I) |
|
Zero matrix |
A matrix with all zero entries |
|
Upper Triangular matrix |
Square matrix having all the entries zero below the principal diagonal |
|
Lower Triangular matrix |
Square matrix having all the entries zero above the principal diagonal |
Square matrix
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)
Null Matrix (Zero Matrix)
A square matrix where all elements equal zero
Properties of zero matrices
So, the identity matrix multiplied by any matrix lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!
Trace: the sum of the diagonal of a square matrix.
For example
A × B = C (2 × 3) × (3 × 2) = (2 × 2)
B × A = D (3 × 2) × (2 × 3) = (3 × 3)
A matrix is a rectangular array of different numbers. we can do different operations like addition, multiplication or subtraction.
Matrix size is the number of rows and columns of matrix. the matrix with (m rows) and (n columns) is called m × n. the m & n are calles its dimensions. for example if matrix (A) has 4 rows and 3 columns. we said A is 4 × 3 matrix
The sum A+B of two m-by-n matrices A and B is calculated entrywise:
Pour ajouter deux matrices, il est nécessaire que le nombre de lignes dans la première matrice soit égal au nombre de lignes dans la deuxième matrice et le nombre de colonnes dans la première matrice est égal au nombre de colonnes dans la deuxième matrice.
The product cA of a number c and a matrix A is computed by multiplying every entry of A by c:
The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
La condition est la même que dans l'addition de matrice. il est nécessaire que le nombre de lignes dans la première matrice soit égal au nombre de lignes dans la deuxième matrice et le nombre de colonnes dans la première matrice est égal au nombre de colonnes dans la deuxième matrice.
Pour multiplier deux matrices, le nombre de colonnes de la première matrice doit être égal au nombre de lignes de la deuxième matrice.