Matrices
A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular array of entries are enclosed in an ordinary bracket or in square bracket.
Types of Matrices
Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity or Unit Matrix, Null Matrix or Zero Matrix
Operations on Matrices
Equality of Matrices, Addition of Matrices, Matrix Addition is commutative, Matrix addition is associative, Subtraction of Matrices, Multiplication of a matrix by a scalar, Multiplication of Matrices, Properties of Matrix Multiplication, Transpose of a Matrix, Properties of Transpose, Symmetric Matrix, Skew-Symmetric Matrix, Properties of Symmetric and Skew Symmetric Matrices,
Determinants
Let A = [aij] be a square matrix. We can associate with the square matrix A, a determinant which is formed by exactly the same array of elements of the matrix A. A determinant formed by the same array of elements of the square matrix A is called the determinant of the square matrix A and is denoted by the symbol det.A or |A|.
Adjoint and Inverse of a Matrix
The adjoint of a square matrix [aij] is defined as the transpose of the matrix [Aij] where Aij are the cofactors of the elements aij. Adjoint of A is denoted by adj A.
Let A be a square matrix of order n. If there exists a matrix B of order n such that AB = BA = I, where I is the identity matrix of order n, then the matrix A is said to be invertible and B is called the inverse (or reciprocal) of A.
A square matrix A is said to be non-singular if its determinant value is non-zero.
A square matrix A is said to be singular if |A| = 0.
Application of Matrices and Determinants
Application of Determinants, Area of a Triangle, Cramer's rule for the solution of a system of equations in 2 variables, Consistency of a system of linear equation.
Application of Matrices, Homogeneous Equations (Constant = 0), Non Homogenous Equations (Solution by the Matrix Method)
Summary
A matrix is defined as a rectangular array of elements. If the arrangement has m rows and n columns, then the matrix is of order mxn (read as m by n). A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter.
Conclusion
In this chapter, we have seen how arranging numbers in orderly rows and columns under the guise of Matrices and Determinants, has helped to solve linear equations or find the area of a triangle. There are in fact other much wider applications in Science and Engineering and other fields.
