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Matrices

A departmental store maintains its monthly sales details in a spread sheet. The rows are used to represent the items and the sales amount for the items are shown in columns representing the day in the month. Such arrangement of collective information in rows and columns is called a matrix.

The plural for matrix is matrices.

Definition of a Matrix:


Matrix is a rectangular array consisting of numbers arranged in rows and columns. If a matrix contains "m" rows and "n" columns, it is called an m $\times$ n matrix (read as m by n) or a matrix of size m $\times$ n.

Example:

$A_{4\times 3} = \begin{bmatrix}
2 &-1  & \frac{3}{2}\\
1 & 0 & -2\\
\frac{2}{5} & -3 &1 \\
2 & -1 & 0
\end{bmatrix}$

Matrix A in the example given has 4 rows and 3 columns and hence a 4 $\times$ 3 matrix. Each entry in the matrix is called an element of the matrix.

The general entry of matrix is written as $a_{ij}$, where i and j correspondingly represent the row and column the element is contained in. Thus, $a_{11}$ = 2, $a_{12}$ = -1, $a_{13}$ = $\frac{3}{2}$, $a_{21}$ = 1, $a_{22}$ = 0, $a_{23}$ = -2 and so on.

The notation {$a_{ij}$} containing the general element is also used to represent the matrix $A_{ij}$.

Although, matrices are widely used in solving many mathematical problems. Their most significant application is perhaps seen in solving systems of linear equations.

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Types of Matrices

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There are four different types of matrix as follows:
  • Square matrix
  • Diagonal matrix
  • Unit matrix
  • Zero matrix
Square Matrix:

A matrix of size n $\times$ n, where number of rows = number of columns = n is called a Square Matrix of order n. For a square matrix, the entries $a_{11}$, $a_{22}$, $a_{33}$, ...., $a_{nn}$ form the main diagonal of the matrix.

Example:

A 3 x 3 square matrix is given below.
Square Matrix
The entries 1, 2 and 0 form the main diagonal of the matrix and are also called the diagonal elements.

Diagonal Matrix:

A diagonal matrix is a square matrix, whose non diagonal entries are all zero. In general, B = {$b_{ij}$} is a Diagonal Matrix, where $b_{ij}$ = 0, whenever i $\neq $ j.

Example:

$B_{44}= \begin{bmatrix}
2 &0  & 0 &0 \\
0 & 3 & 0 & 0\\
0 & 0 & -1 &0 \\
0 & 0& 0& 1
\end{bmatrix}$

Unit or Identity Matrix:

A unit or identity matrix is a diagonal matrix, whose diagonal elements are all 1's and remaining are 0's.

Identity Matrix of order 2 is as follows:

$\begin{bmatrix}
1 & 0\\
 0& 1
\end{bmatrix}$
Identity Matrix of order 3
$\begin{bmatrix}
1 & 0 &0 \\
0 & 1 &0 \\
0 & 0 & 1
\end{bmatrix}$

Identity matrix get their name from the fact that they serve as identity in matrix multiplication.

Zero Matrix:

If the elements of a matrix are all zeros, then it is called a zero matrix.

Examples:

$0_{23}= \begin{bmatrix}
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}$
    
$0_{23}= \begin{bmatrix}
0 & 0 & 0\\
 0& 0 & 0\\
 0&  0& 0
\end{bmatrix}$

Zero matrix of order m x n serves as the additive identity Matrices of m x n size.

More topics in  Matrices
Properties of Matrices Types of Matrices
Matrix Operations Transpose of a Matrix
Determinant of a Matrix Augmented Matrix
Augmented Matrices Squaring Matrices
Adjoint Matrix Eigenvalues and Eigenvectors
Matrix Decomposition
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