Matrices

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

For 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.

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

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.

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,

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:

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:

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.

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:

Properties of an Identity Matrix

Triangular Matrices

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

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A square matrix having zeros at all positions below the main diagonal.

A square matrix having zeros at all positions above the main diagonal.

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Transpose Matrix

Transpose of matrix A is denoted by A^T

Two rows of A^T are the columns of A.
The columns of A^T are rows of A.

If A is m x n matrix then, A^T is n x m matrix.

Symmetric Matrices

A is a square matrix.
If A = A^T, A is Symmetric Matrix

Let’s see an example.

Now take the transpose of A.

We can see that,

A = A^T

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

Example 02

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:

Find 4A,

Multiplication of a Matrix by Another Matrix

It is easier to learn through an example.

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

Let’s take,

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

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

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

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

Let’s see another example.

PQ will be 3 x 2 matrix,

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:

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:

Reduce the following matrix to the echelon form.

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FAQ

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