TYPES OF MATRICES AND DETERMINANTS
LEARNING OBJECTIVES:
1. Identify different types of matrices, including square matrices, rectangular matrices,
symmetric matrices, skew-symmetric matrices, diagonal matrices, and identity matrices.
2. Understand the properties and characteristics of each type of matrix, including their
definitions and representations.
3. Define determinants of square matrices and comprehend their significance in determining
invertibility, consistency, and volume scaling in linear transformations.
4. Learn the properties of determinants, including their behavior under elementary row
operations, transpose, and matrix inversion.
5. Apply properties of determinants to compute determinants of matrices using various
methods, such as cofactor expansion, row reduction, and properties of special matrices.
6. Solve problems involving determinants, including calculating the determinant of a matrix,
determining the rank of a matrix, and solving systems of linear equations using Cramer's
Rule or determinants.
Types of Matrices
Types of Matrices classifies matrices in different categories based on the number of rows and
columns present in them, the position of the elements, and also the special properties exhibited by
the Matrix. Matrix is a rectangular array of numbers in which elements are arranged in rows and
columns. Each element is identified as aij where i and j indicate the row and column number
respectively for the element. We have different types of matrices which are classified on the basis
of the number of rows and columns, the elements present in them, the order of the matrix, and the
properties shown by the matrix. In this article, we will learn about different types of matrices in
detail along with a brief introduction to matrices.
Row Matrix
A matrix having only one row is called a row matrix. Thus A = [aij]mxn is a row matrix if m = 1.
So, a row matrix can be represented as A = [aij]1×n. It is called so because it has only one row, and
the order of a row matrix will hence be 1 × n. For example, A = [1 2 4 5] is a row matrix of order
1 x 4. Another example of the row matrix is P = [ -4 -21 -17 ] which is of the order 1×3.
Column Matrix
A matrix having only one column is called a column matrix. Thus, A = [aij]mxn is a column matrix
if n = 1. So, the value of a column matrix will be 1. Hence, the order is m × 1.
Zero or Null Matrix
If all the elements are zero in a matrix, then it is called a zero matrix and generally denoted by 0.
Thus, A = [aij]mxn is a zero-matrix if aij = 0 for all i and j; E.g.
Singleton Matrix
If there is only one element in a matrix, it is called a singleton matrix. Thus, A = [aij]mxn is a
singleton matrix if m = n = 1. E.g. [2], [3], [a], [] are singleton matrices.
Diagonal Matrix
If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a diagonal
matrix. Thus, a square matrix A = [aij] is a diagonal matrix if aij = 0,when i ≠ j.
Conclusions:
• All identity matrices are scalar matrices
• All scalar matrices are diagonal matrices
• All diagonal matrices are square matrices
It should be noted that the converse of the above statements is not true for any
of the cases.
Triangular Matrix
A square matrix is said to be a triangular matrix if the elements above or below the principal
diagonal are zero, and there are of two types:
Singular Matrix and Non-Singular Matrix
Matrix A is said to be a singular matrix if it’s determinant |A| = 0; otherwise, a non-singular matrix,
i.e. if for det |A| = 0, it is singular matrix and for det |A| ≠ 0, it is non-singular.
ACTIVITY NO.3
Answer the following questions. Explain and give at least 1 illustration for each.
1. What do you mean by a symmetric matrix?
2. What do you mean by identity matrix?
3. What do you mean by Hermitian matrix?