# 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 likeA, B … ZVectors are denoted by bold small case letters like alphabets likea,b…z

**Null Matrix** Null matrix is a matrix all elements

**0**s. 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

**1**s on all the main diagonal positions and

**0**s 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