matrices

Matrices

Matrices are, a rectangular block of numbers arranged in to rows and columns.

For an example,

matrices example

There are some unique terms that we should know when we are dealing with matrices.

Dimensions of Matrix

When we consider the above example it has two rows and three columns. So, the dimensions of matrix A is 2 x 3

Dimension of a matrix = Number of rows x Number of columns

Let’s find the dimension of the following matrices.

matrices example 2x2

Dimension of A = 2 x 2
We called this one as two by two matrix.

matrices example 4x2

Dimension of C = 4 x 2
We called this one as four by two matrix.

Matrix Elements

Entries in a matrix are called elements of a matrix. Elements are defined by using rows and columns.

Let’s see an example.

matrix element x

Equal Matrices

If matrix A = matrix B we can say that A and B are identical.

To A = B

01) The matrix A and B should be the same size.
02) Corresponding elements should be equal.

If,

equal matrices

Then A = B means,

b11=1 , b12=2 , b13=3 , b21=4 , b22=5 , b23=6 , b31=7 , b32=8 , b33=9

Square Matrices

If the number of rows and columns of a matrix are same they are called Square Matrices.

Example:

3x3 matrix

2×2 , 3×3 , 4×4 , 5×5 , 6×6 , … matrices are examples for square matrices.

Zero Matrix

A matrix which consist of 0 s is called a Zero Matrix.

Examples:

zero matrix

Properties of a Zero Matrix

Properties of a Zero Matrix
Properties of a Zero Matrix
Properties of a Zero Matrix
Properties of a Zero Matrix

(Addition and multiplication of matrices will be describe later in this article.)

Diagonal Matrix

A diagonal matrix has zero entries all over the matrix except in the main diagonal. Diagonal matrices always come under square matrices.

diagonal matrix
diagonal matrix

Identity Matrix

Identity Matrix is a matrix that has 1 s as the entries in the main diagonal.

We indicate identity matrices usually by the letter I

Examples:

identity matrix

Properties of an Identity Matrix

identity matrix properties
identity matrix properties
identity matrix properties
identity matrix properties

Triangular Matrices

The main diagonal divides a square matrix in to two triangles.

triangular matrices

types of triangular matrices
A square matrix having zeros at all positions below the main diagonal.A square matrix having zeros at all positions above the main diagonal.
triangular matrices types

Transpose Matrix

Transpose of matrix A is denoted by AT

Two rows of AT are the columns of A.
The columns of AT are rows of A.

If A is m x n matrix then, AT is n x m matrix.

transpose matrix
transpose matrix

Symmetric Matrices

A is a square matrix.
If A = AT, A is Symmetric Matrix

Let’s see an example.

symmetric matrices

Now take the transpose of A.

properties of symmetric matrices

We can see that,

A = AT

So A is a Symmetric Matrix.

Before learning other definitions we have to learn about the addition and multiplication of matrices.

Addition of Matrices

If A and B are two matrices of the same size, we can get a matrix for A + B by adding the corresponding elements of A and B

Example 01

addition of matrices
addition of matrices
matrices addition

Example 02

matrices addition example

Multiplying Matrices

Multiplication of a Matrix by a Number

If A is a matrix and k is any real number, we can find kA by multiplying each element of matrix A by k.

Example:

multiplying matrices

Find 4A,

multiplying matrices
multiplying matrices

Multiplication of a Matrix by Another Matrix

It is easier to learn through an example.

multiply matrix by matrix

A is a 2 x 3 matrix, B is a 3 x 2 matrix.
AB will be,

multiply matrix by matrix order

Let’s take,

simple matrix multiplication

(Element in 1st row 1st column)
g11 = ( 2 x 6 ) + ( 4 x 0 ) + ( 3 x -3 ) ; Multiply the 1st row entries of A by 1st column entries of B.
= 12 + 0 – 9
= 3

(1st row 2nd column)
g12 = ( 2 x 2 ) + ( 4 x 5 ) + ( 3 x 1 )
= 4 + 20 + 3
= 27

(2nd row 1st column)
g21 = ( 1 x 6 ) + ( 5 x 0 ) + ( 6 x -3 )
= 6 + 0 – 18
= -12

(2nd row 2nd column)
g22 = ( 1 x 2 ) + ( 5 x 5 ) + ( 6 + 1 )
= 2 + 25 + 6
= 33

simple matrix multiplication

Let’s see another example.

multiply 3x3 matrix by 3x2

PQ will be 3 x 2 matrix,

multiply 3x3 matrix by 3x2
multiplying 2 matrices
multiplying 2 matrices
multiplying 2 matrices

We can see that when we multiply a matrix by an identity matrix it will always give the same matrix.

Echelon Form of a Matrix

A matrix is said to be in Echelon form if,

a) All non-zero rows are above any rows of all zeros.
b) The leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
c) The number of zeros proceeding the first nonzero element of a row increases as we proceed from row to row downwards.

Eg:

Echelon Form of a Matrix

Row – Reduced Echelon Form of a Matrix

A matrix is said to be in row reduced echelon form when it satisfies the following properties.

a) The first nonzero entry in each row is 1.
b) Each successive row has its first nonzero entry in a later column.
c) All entries (above and) below the first nonzero entry of each row are zero.
d) All full rows of zeros are the final rows of the matrix.

Eg:

row reduced echelon form

Reduce the following matrix to the echelon form.

row reduced echelon form
row reduced echelon form 2
row reduced echelon form example 3
row reduced echelon form example 4

FAQ

What are matrices?

Matrices are, a rectangular block of numbers arranged into rows and columns.matrices example

What are the dimensions of a matrix?

If we consider this image, the dimensions of this matrix A is 2 x 3.
The dimension of a matrix = Number of rows x number of columns
matrices example

What are the Matrix Elements?

Entries in a matrix are called elements of a matrix. Elements are defined by using rows and columns.matrix element x

What are the Equal Matrices?

If matrix A = matrix B we can say that A and B are identical.
To A = B

What are the square matrices?

If the number of rows and columns of a matrix are same they are called Square Matrices.

What is a Zero Matrix?

A matrix which consist of 0 s is called a Zero Matrix.

What is a Diagonal Matrix?

A diagonal matrix has zero entries all over the matrix except in the main diagonal. Diagonal matrices always come under square matrices.diagonal matrix

What is Identity matrix?

Identity Matrix is a matrix that has 1 s as the entries in the main diagonal.

What are the Triangular Matrices?

The main diagonal divides a square matrix in to two triangles.types of triangular matricestriangular matrices

What is Echelon Form of a Matrix?

A matrix is said to be in Echelon form if,
a) All non-zero rows are above any rows of all zeros.
b) The leading coefficient of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
c) The number of zeros proceeding the first nonzero element of a row increases as we proceed from row to row downwards.

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