In the previous chapter, we discussed Introduction of Matrix. Here, we shall discuss Types of Matrix.
Row Matrix
Row Matrix is that type of matrix in which the entries are arranged in only one row and more than one columns. The general order of a row matrix is 1xn where n = 1,2,3,4,… .
Examples of Row Matrix:
$A = [\displaylines{1 & 2}]_{1×2}$
$A = [\displaylines{1 & 2 & 3}]_{1×3}$
Column Matrix
Column Matrix is that type of matrix in which entries are arranged in only one columns and more than one rows. The general order of a column matrix is mx1 where m = 1,2,3,4,…. .
Examples of Column Matrix:
$A = \left [ \displaylines{x \\ y} \right ]_{2×1}$
$A = \left [ \displaylines{a \\ b \\ c} \right ]_{3×1}$
Rectangular Matrix
A Rectangular Matrix is that type of matrix in which data are arranged in unequal number of rows and columns. Row and Column matrices are also the examples of rectangular matrix. The general order of a rectangular matrix is mxn where m,n = 1,2,3,… and m $\neq$ n.
Examples of Rectangular Matrix:
$\text{B} = \left [ \displaylines{a & b & c \\ d & e & f} \right]_{2×3}$
$\text{C} = \left [ \displaylines{a & b \\ c & d \\ e & f} \right]_{3×2}$
Square Matrix
A Square Matrix is that type of matrix in which data are arranged in equal number of rows and columns. The general order of a square matrix is mxm or nxn where m,n = 1,2,3,… .
Examples of Square Matrix:
$\text{B} = \left [ \displaylines{a & b \\ c & d} \right]_{2×2}$
$\text{C} = \left [ \displaylines{a & b & c \\ d & e & f \\ g & h & i} \right]_{3×3}$
Types of Square Matrices
Null or Zero Matrix
A Null matrix is that type of squarematrix, either square or rectangular, having all of its elements equal to zero. The general order of null matrix is mxm or nxn where m,n = 1,2,3,… .
Examples of Zero Matrix:
$\text{O} = \left [ \displaylines{0 & 0 \\ 0 & 0} \right]_{2×2}$
$\text{O} = \left [ \displaylines{0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0} \right]_{3×3}$
Unit or Identity Matrix
A unit matrix is a square matrix whose all elements of the principal diagonal are unit or 1. Identity matrices are represented by the letter I with a number at its base as $\text{I}_2$. The base number represents the order of the matrix.
Example of Unit Matrix:
$\text{I}_2 = \left [ \displaylines{1 & 0 \\ 0 & 1} \right ]_{2×2}$
$\text{I}_2 = \left [ \displaylines{1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1} \right ]_{3×3}$
Diagonal Matrix
Diagonal Matrix is a type of square matrix whose elements other than the principal diagonal are equal to zero.
Examples of Diagonal Matrix:
$\text{A} = \left [ \displaylines{2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 7} \right ]_{3×3}$
Scalar Matrix
Scalar Matrix is a square matrix whose all elements on the principal diagonal are the same quantity. In other words, a square matrix that satisfies aij = k for all values of i = j, where k is any real number is said to be a scalar matrix.
Examples of Scalar Matrix:
$\text{A} = \left [ \displaylines{2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2} \right ]_{3×3}$
Symmetric Matrix
Symmetric Matrix is a square matrix of order mxn which holds the relation aij = aji true for all values of i and j.
Examples of Symmetric Matrix:
$\text{A} = \left [ \displaylines{1 & 4 \\ 4 & 2} \right ]_{2×2}$
$\text{A} = \left [ \displaylines{1 & 4 & 6 \\ 4 & 2 & 8 \\ 6 & 8 & 3} \right ]_{3×3}$
Skew-symmetric Matrix
Scalar Matrix is a square matrix of order mxn which holds the relation aij = -aji true for all values of i and j.
Examples of Skew-symmetric Matrix:
$\text{A} = \left [ \displaylines{0 & 4 \\ -4 & 0} \right ]_{2×2}$
$\text{A} = \left [ \displaylines{0 & 4 & 6 \\ -4 & 0 & -8 \\ -6 & 8 & 0} \right ]_{3×3}$
Triangular Matrix
Triangular matrix is a square matrix whose elements either above the principal diagonal or below the principal diagonal are zero. If the elements above the principal diagonal are zero, the matrix is said to be lower triangular matrix. Otherwise, if the elements below the principal diagonal are zero, the matrix is said to be upper triangular matrix.
Examples of Triangular Matrix:
Upper Triangular Matrix:
$\text{A} = \left [ \displaylines{5 & 4 & 6 \\ 0 & 5 & 8 \\ 0 & 0 & 3} \right ]_{3×3}$
Lower Triangular Matrix:
$\text{A} = \left [ \displaylines{5 & 0 & 0 \\ 5 & 4 & 0 \\ 3 & 5 & 9} \right ]_{3×3}$
Equal Matrix
Two or more matrices of the same order are said to be equal matrix if their corresponding elements are equal.
For example: If A and B be two matrices defined as:
$\text{A} = \left [ \displaylines{ a_{11} & a_{12} \\ a_{21} & a_{22} } \right ]$ and $\text{B} = \left [ \displaylines{ b_{11} & b_{12} \\ b_{21} & b_{22} } \right ]$
Matrices A and B are equal if they satisfy the following relations:
- $a_{11} = b_{11}$
- $a_{12} = b_{12}$
- $a_{21} = b_{21}$
- $a_{22} = b_{22}$
- $a_{mn} = b_{mn}$