Matrix is the most powerful tool in mathematics. We use matrices in certain branches of science as far as in economics, computer, psychology and management.

A matrix is an arrangement of numbers or symbols in rectangle array. The items in a matrix are called elements of matrix. Given below is an example of a matrix with eight elements:

1 &3 &4 & -1 \\
2 &5 &-3 & 5

With the help of matrices, we can solve a system of linear equation.

A matrix is a rectangular arrangement of numbers. We arrange numbers in rows(horizontal lines) and columns(vertical lines). In general, a matrix with m rows and n column is denoted by $A_{m\times n }$ and called $m\times n$ order matrix.

Order of a matrix (also called as the dimension of the matrix) represents the number of rows and column in a given matrix. So, $A_{m\times n }$ =$\left [c_{i,j}Â \right ]_{m\times n }$, where $c_{i,j}$ refers to the elements of matrix and subscript m and n refers to the row number and column number respectively.

The element $c_{i,j}$ belongs to the ith row and jthcolumn and is sometimes called the (i,j)th element of the matrix. Here i = 1, 2..... m and j = 1, 2, 3..... n.


In matrix, the number of columns and rows need not to be equal.
In mathematics, there are different types of matrices which are classified according to their order as follows:

  • Row Matrix
  • Column Matrix
  • Square Matrix
  • Diagonal Matrix
  • Symmetric and Skew-symmetric Matrix
  • Null Matrix
  • Unit Matrix

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Let A and B be two matrices of same order. These matrices are said to be equal if their corresponding elements are equal.

Let $A_{m\times n}$ = [ xij ] and $B_{m\times n}$ = [ yij ] are two matrices of the same order ($m\times n$) if $ x_{ij} = y_{ij}$ for all i and j. So, we write A = B.

Solved Examples

Question 1: A = $\begin{bmatrix}
1 &2 \\
1 &2
\end{bmatrix}$ and B = $\begin{bmatrix}
1 &2 \\
1 &2
Find whether the matrices A ad B are equal.

From the above example, it is clear that matrices A and B are of same order $(2\times 2)$ and their corresponding elements are equal. So, A and B are equal matrices.

Question 2: Find whether the following matrices are equal matrices. C = $\begin{bmatrix}
1 &3 &5 &7
\end{bmatrix}$ and D = $\begin{bmatrix}
8 &1 \\
2 &3 \\
4 &5
Here, order of C1x4 is not equal to the order of D2x3. So, C$\neq$D. By the use of the above property, we can find out the value of unknown variables.

Question 3: Find the value of x and y in the matrices A =$\begin{bmatrix}
x &3 &6 \\
1 &y-2 &9
\end{bmatrix}$ and B =$\begin{bmatrix}
2 &3 &6 \\
1 &1 &9
It is clear from the above example that both matrices have the same order $(2\times3)$. So, we use the property which states that corresponding elements should be equal.
i.e. x = 2, 3 = 3, 6 = 6, 1 = 1, y - 2 = 1, 9 = 9.
So, we get x = 2 and y = 3.

Just like operations on numbers, we use some operations on matrix also. They are addition, subtraction, multiplication etc.

Matrix Addition and Subtraction: Matrix addition and subtraction are possible only if the order of two matrices A & B are the same. We can add or subtract the corresponding elements of the matrices.
i.e. A $\pm$ B = [aij] $\pm$ [bij]

Matrix Multiplication: If $A_{p\times q}$ = [aij] and $B_{q\times r}$ = [bij] are two matrices, then AB is a matrix of order $p\times r$.
So, $(AB)_{p\times r}$ = [cij]
Let A = [aij] be a given matrix. If we interchange the rows and columns of matrix A, then the resultant matrix is known as transpose of matrix A and is denoted by AT, i.e. If A = [aij], then AT = [aij].

Solved Example

Question: Find the transpose of the matrix $A_{3\times 3}$=$\begin{bmatrix}
a &b &-c \\
-d &e &s \\
u &v &w
a &-d &u \\
b &e &v \\
-c &s &w

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If A = [aij] is a square matrix, then the determinant whose elements are aij is called the determinant of A and is denoted by $\left | A \right |$.
Let A = $\begin{bmatrix}
a_{11} &a_{12} \\
a_{21} &a_{22}
The determinant of A is as follows:
$\left | A \right |$ = (a11a22 - a21a12). → Read More
If A = [aij] is a square matrix, then to determine adjoint of matrix A, first we have to calculate the cofactor matrix of A and then the transpose matrix of that cofactor matrix. Thus, the resultant matrix is known as Adjoint matrix of the given matrix and is denoted by adj A.
A = [aij] The adjoint of matrix A is as follows:
adj A=$\left [c _{ij} \right ]^{T}$ → Read More
If A be the given square matrix, then the inverse of A is written as A-1 and is defined as follows:
A -1 =$\frac{adjA}{\left | A \right |}$, here $\left | A \right | \neq 0$
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In mathematics, there are four basic operations on numbers namely addition, subtraction, multiplication and division. But in matrices,there are only three operation that we can apply on rows of a given matrix:
  • Interchanging the two rows like $R_{1}\leftrightarrow R_{2}$
  • Multiply any row by a scalar or number $3R_{1}$ or $4R_{2}$
  • Multiply k times the elements of a row. Then, add and subtract p times the corresponding elements of another row, where k and p are real numbers.
These operations are called Elementary Row Transformation. We can easily understand these operations from the examples shown below:

Let A = $\begin{bmatrix}
2 &1 &3 &7 \\
2 &6 &4 &8 \\
1 &1 &0 &1

Apply the following operation on matrix A, $R_{1}\leftrightarrow R_{3}$ (interchanging the rows)

B $\approx $ $\begin{bmatrix}
1 &1 &0 &1 \\
2 &6 &4 &8 \\
2 &1 &3 &7

Apply the following operation on matrix B, $R_{2}\rightarrow$$\frac{1}{2}\times R_{2}$ (multiplying row by a number)

C $\approx $ $\begin{bmatrix}
1 &1 &0 &1 \\
1 &3 &2 &4 \\
2 &1 &3 &7

Apply the following operation on matrix C, $R_{3}\rightarrow R_{3}-2R_{2}$ (operation between two rows)

D $\approx $ $\begin{bmatrix}
1 &1 &0 &1 \\
1 &3 &2 &4 \\
0 &-5 &-1 &-1
Thus, after applying Elementary Transformation on A, we have obtained the matrix D.
Listed below are some of the properties of matrices.

Properties of addition: Let A and B are two matrices of the same order ($m\times n$) and c be any scalar.
Commutative Property
A + B = B + A
Associative Property (A + B) + C = A + (B + C)
Existence of an Additive Identity B + 0 = B = 0 + B
Existence of an Additive Inverse B + (-B) = 0
Distributive Property c(A + B) = cA + cB

Properties of multiplication: Let $A_{l \times m}$, $B_{m \times n}$, $C_{n \times p}$ are three matrices and t,q are the scalars.

Associative property
A(BC) = (AB)C
Distributive property A(B + C) = AB + BC
Multiplicative Identity 1A = A
Property for scalar multiplication (tq)A = t(qA)
Existence of Multiplicative Inverse A (A-1) = In

Properties of Transpose:
  • (AB)T = BTAT
  • (A + B)T = AT + BT
  • (AT)T = A
  • (qA)T = qAT

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