Vector/Matrix Algebra

What is the Matrix?

Matrix
The ordered rectangular array

Is called a Matrix of order m*n
where i=1,2,3… m and j=1,2,3… n.

Vector
A vector is a matrix with only one column.

Note:
Multicolumn matrices are denoted by bold uppercase alphabets like A, B … Z
Vectors are denoted by bold small case letters like alphabets like a, b… z

Null Matrix
Null matrix is a matrix all elements 0s. It’s denoted by Z and defined as

Equal Matrix
Let

And

Then A=B if and only if they are of same order and all corresponding elements are also equal i.e. n=q and m=p and

Identity Matrix
Identity matrix is a square matrix with 1s on all the main diagonal positions and 0s every where else. It’s denoted by I and defined as

Ex:

Matrix Addition
The addition of two matrix is only defined if they are of same order.
Let
A(m * n) & B(m * n)
Then it’s summation C = A + B is defined as

For all i = 1, 2, 3 … m j = 1, 2, 3 … n
C’ll be m * n matrix

Matrix Scalar Multiplication
Let

Then their product is

is a matrix B of order m*n whose elements are defined as

Matrix Multiplication
Two matrices are multiplicable if and only if they are of orders A(m*n) and B(n*p).

And it’s product C = AB is defined as

For all i = 1, 2, 3 … m j = 1, 2, 3 … p.
C’ll be m * p matrix

Matrix Transpose
Transpose of a matrix A is denoted as

and is defined as a matrix whose elements are given by eq

Matrix Trace
Trace of a matrix is defined only for square matrix. Let A is (n*n). Then trace of the matrix A, tr(A) is defined as sum of all the elements on it’s main diagonal.

SubMatrix
Let A is m*n matrix. Then the sub matrix is (m-1)*(n-1) matrix obtained from A by removing row i and column j.
Ex

Then sub matrix

Matrix Determinant
Determinant of a matrix is only defined for square matrix.
Let A is square matrix
Then determinant of A is denoted as det(A) or |A| and is defined as

Cofactor Matrix
Cofactor matrix is defined for only square matrix.
Let A is n*n matrix. Then cofactor matrix C of A is defined as

Adjugate of a Matrix
Adjugate of a matrix is defined only for square matrix.
Let A is n*n matrix. Then Adjugate of A is denoted by adj(A) and is defined as.

Inverse of Matrix
Inverse of a matrix is defined for only for square matrix.
Let A is (n*n) matrix
Inverse of a matrix is denoted as

And is defined as

Where I is an identity matrix of order n. A is invertible if and only if

And is evaluated by the equation