Determination of Order of Matrices: Definition, Types, Solved Problems

Matrices are one of the most important and most powerful tools in mathematics. Matrices help us to simplify our work to a greater extent when compared to other methods. To obtain compact and simple solutions for linear equations, the concept of the matrix has evolved. It is used to represent coefficients in the system of linear equations, it is also used in different areas of business such as budgeting, cost estimation, analysis of the result of an experiment or to determine sales projection. This mathematical tool is also used in different branches of science, genetics, economics, sociology, industrial management, cryptography and modern physics. 

Also Read:  Algebra

Key Terms: Matrices, Matrix, Rows, Columns, Order of Matrix, Elements, Square Matrix, Determinants, Linear equations, Array

---

Matrix

[Click Here for Sample Questions]

Matrices are defined as numbers of functions in a rectangular array. It is a 2- dimensional function because of the rectangular array. Two-dimensional matrices consist of rows and columns. Rows and columns are donated by ‘m’ and ‘n’ respectively. 

A matrix is a rectangular array of symbols or numbers arranged in the form of rows and columns. The plural form of a matrix is matrices. A number of rows and columns are known as the order of the matrix, and each number in the matrix is known as an element. The order of the matrix is given by ‘m x n’, where ‘m’ and ‘n’ are the number rows and columns respectively.

The video below explains this:

Matrices Detailed Video Explanation:

Also Read: Elimination Method of Solving a Pair of Linear Equations 

---

Determination of Order of Matrix

[Click Here for Sample Questions]

Order of the matrix is determined by the information of a number of rows and columns, basically, it provides dimensions of the matrix. It is mostly represented by A_{m x n}, where ‘m’ is the number of rows and ‘n’ is the number of columns.

Order of matrix

The general notation of the matrix is given by:

A = [a_ij]_{m x n},

Where, 1\(\leq\) i\(\leq\) m,

1\(\leq\) j \(\leq\)n,

And, i,j \(\in\)N

It is important to note that the matrix is denoted by a capital letter and the elements are denoted by the same letter but in a small case.

Therefore, a_ij represents any element of the matrix which is in i^th row and j^th column.

Example: Consider two matrices A and B

 \(A = \begin{bmatrix} 3 & 4 & 5 \\[0.3em] 1 & 8 &9 \\[0.3em] -4 & 9 & 25 \\[0.3em] 0 & 16 & -30 \end{bmatrix}\)

\(B = \begin{bmatrix} 1 & 20 \\[0.2em] 10 & 5 \\[0.2em] \end{bmatrix}\)

For A,

There are four rows and three columns. Therefore the order of the matrix is [A]_{4 x 3}

That is, 

\(A = \begin{bmatrix} 3 & 4 & 5 \\[0.3em] 1 & 8 &9 \\[0.3em] -4 & 9 & 25 \\[0.3em] 0 & 16 & -30 \end{bmatrix}_{4 X 3}\)

Elements,

For example, a₄₂= 16

In the fourth row, i = 4 and in the second column j= 2, so the value(element) at the fourth row and the second column are 16.

Similarly, a₁₃= 5, and so on,

For B,

There are two rows and two columns. Therefore the order of the matrix is [B]_{2 x 2}

That is,

\(B = \begin{bmatrix} 1 & 20 \\[0.2em] 10 & 5 \\[0.2em] \end{bmatrix}_{ 2 X 2}\)

Notation for 5 in B matrix is b₂₂.

Number of elements

The number of elements in the matrix is the multiplication of order of matrix

If, \(A = \begin{bmatrix} 3 & 4 & 5 \\[0.3em] 1 & 8 &9 \\[0.3em] -4 & 9 & 25 \\[0.3em] 0 & 16 & -30 \end{bmatrix}_{4 X 3}\)

Total number of elements is 4 x 3 =12

If \(B = \begin{bmatrix} 1 & 20 \\[0.2em] 10 & 5 \\[0.2em] \end{bmatrix}_{2 X 2}\) 

Total number of elements is 2 x 2 = 4

Also Read: Matrix multiplication

---

Different Types of Matrices

[Click Here for Sample Questions]

• Row Matrix

Number of rows in this matrix is one, and it is fixed. The number of columns can vary.

For example, \(\begin{bmatrix} 1 & 13 & 14 & 5 & 7 \\[0.2em] \end{bmatrix}_{1 X 5}\)

• Column Matrix

Number of columns in this matrix is one, and it is fixed. The number of rows can vary.

Example, \(\begin{bmatrix} 14 \\[0.2em] 5 \\[0.2em] 91 \\[0.2em] \end{bmatrix}_{3 X 1}\) 

• Singleton Matrix

In this matrix the number of rows and number of columns is one, and it's fixed. This means that the number of elements in this matrix will always be one.

Example, \(\begin{bmatrix} 3 \end{bmatrix}_{1 X 1}\)

• Rectangular Matrix

In this matrix the number of rows and number of columns are different. They are never the same.

Example, \(\begin{bmatrix} 3 & 4 & 5 \\[0.3em] 1 & 8 &9 \\[0.3em] -4 & 9 & 25 \\[0.3em] 0 & 16 & -30 \end{bmatrix}_{4 X 3}\), \( \begin{bmatrix} 13 & 1 & 8 \\[0.2em] 18 & 5 & 9 \\[0.2em] \end{bmatrix}_{2 X 3}\)

• Square Matrix

In this matrix the number of rows and number of columns are the same.

Example, \(\begin{bmatrix} 5 & 4 & 6 \\[0.3em] 8 & 3 &2 \\[0.3em] 4 & 6 & 4 \\[0.3em] \end{bmatrix}_{3 X 3}\), \( \begin{bmatrix} 1 & 20 \\[0.2em] 10 & 5 \\[0.2em] \end{bmatrix}_{2 X 2}\)

• Null Matrix

All the elements in this type of matrix are zero.

Example, \(\begin{bmatrix} 0 & 0 & 0 \\[0.3em] 0 & 0 & 0 \\[0.3em] 0 & 0 & 0 \\[0.3em] \end{bmatrix}_{3 X 3}\)

• Diagonal Matrix

In this matrix all the elements are zero, except the diagonal elements.

Example, \(\begin{bmatrix} 1 & 0 & 0 \\[0.3em] 0 & 2 & 0 \\[0.3em] 0 & 0 & 6 \\[0.3em] \end{bmatrix}_{3 X 3}\), \(\begin{bmatrix} 0 & 0 & 1 \\[0.3em] 0 & 5 & 0 \\[0.3em] 4 & 0 & 0 \\[0.3em] \end{bmatrix}_{3 X 3}\)

• Scalar Matrix

This is a type of diagonal matrix in which all the diagonal elements are the same.

For example, \(\begin{bmatrix} 2 & 0 & 0 \\[0.3em] 0 & 2 & 0 \\[0.3em] 0 & 0 & 2 \\[0.3em] \end{bmatrix}_{3 X 3}\), \( \begin{bmatrix} 0 & 10 \\[0.2em] 10 &0 \\[0.2em] \end{bmatrix}_{2 X 2}\)

• Identity Matrix

Identity Matrix is a type of Scalar matrix in which all the diagonal elements are 1.

Example, \(\begin{bmatrix} 1 & 0 & 0 \\[0.3em] 0 & 1 & 0 \\[0.3em] 0 & 0 & 1 \\[0.3em] \end{bmatrix}_{3 X 3}\), \( \begin{bmatrix} 0 & 1 \\[0.2em] 1 &0 \\[0.2em] \end{bmatrix}_{2 X 2}\)

• Upper Triangular Matrix

In this type of matrix, in which triangular elements below diagonal elements are zero and the triangular elements above diagonal elements are non-zero.

Example, \(\begin{bmatrix} 1 & 5 & 6 \\[0.3em] 0 & 2 & 4 \\[0.3em] 0 & 0 & 3 \\[0.3em] \end{bmatrix}_{3 X 3}\)

• Lower Triangular Matrix

In this type of matrix, which triangular elements above diagonal elements are zero and the triangular elements below diagonal elements are non-zero.

Example, \(\begin{bmatrix} 1 & 0 & 0 \\[0.3em] 8 & 2 & 0 \\[0.3em] 8 & 6 & 3 \\[0.3em] \end{bmatrix}_{3 X 3}\)

• Symmetric Matrix

In this type of matrix, it has equal values to its transpose A= A^T i.e. a_mn= a_nm. The example of Symmetric matrix is the square matrix.

Example, \(\begin{bmatrix} 2 & 9 & 5 \\[0.3em] 9 & 4 & 1 \\[0.3em] 5 & 1 & 6 \\[0.3em] \end{bmatrix}_{3 X 3}\)

• Anti-symmetric Matrix

In this type of matrix, it has negative values to its transpose A= -A^T, i.e. a_mn = –  a_nm. ‘Skew-symmetric matrix’ is also known as Anti-symmetric matrix.

Example, \(\begin{bmatrix} 1 & -7 & 6 \\[0.3em] 7 & 9 & 4 \\[0.3em] -6 & -4 & 2 \\[0.3em] \end{bmatrix}_{3 X 3}\)

Also Read: Type of Matrices

---

Things to Remember

• A matrix is a rectangular array of numbers or functions.
• They are generally arranged in the form of rows and columns inside the square bracket.
• The number of rows and columns is known as the order of the matrix.
• Elements are the entries are the number in the matrix.
• The size of the matrix is given by ‘m’ and ‘n’.
• ‘m’ and ‘n’ represent the number of rows and columns respectively.

Also Read: