Table of contents 
Introduction 
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A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.
A matrix with 9 elements is shown below.
This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For example, a_{23 }= 6
Order of a Matrix
The order of a matrix is defined in terms of its number of rows and columns.
Order of a matrix = No. of rows × No. of columns
Therefore Matrix [M] is a matrix of order 3 × 3.
Transpose of a Matrix
The transpose [M]^{T }of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].
if A = [a_{ij}] mxn , then AT = [b_{ij}] nxm where b_{ij} = a_{ji}
Properties of transpose of a matrix
Singular and Nonsingular Matrix
Properties of Matrix addition and multiplication
Square Matrix: A square Matrix has as many rows as it has columns. i.e. no of rows = no of columns.
Symmetric matrix: A square matrix is said to be symmetric if the transpose of original matrix is equal to its original matrix. i.e. (A^{T}) = A.
Skewsymmetric: A skewsymmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (A^{T}) = A.
Diagonal Matrix: A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square matrices.
Identity Matrix: A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros. Identity matrix is denoted as I.
Orthogonal Matrix: A matrix is said to be orthogonal if AA^{T} = A^{T}A = I
Idemponent Matrix: A matrix is said to be idemponent if A^{2} = A
Involutary Matrix: A matrix is said to be Involutary if A^{2} = I.
Note: Every Square Matrix can uniquely be expressed as the sum of a symmetric matrix and skewsymmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).
Adjoint of a square matrix: The adjoint of a matrix A is the transpose of the cofactor matrix of A
Properties of Adjoint
Where, “n = number of rows = number of columns”
Inverse of a square matrix
Here A should not be equal to zero, means matrix A should be nonsingular.
Properties of inverse
Where should we use the inverse matrix?
If you have a set of simultaneous equations:
7x + 2y + z = 21
3y – z = 5
3x + 4y – 2x = 1
As we know when AX = B, then X = A^{1}B so we calculate the inverse of A and by multiplying it B, we can get the values of x, y, and z.
Trace of a matrix
Trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix. Remember trace of a matrix is also equal to sum of eigen value of the matrix.
For example:
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