Matrices and Determinants


   
 
Adjoint and Inverse of a Matrix
Adjoint of a Square 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.
 
 
 
                                      
 
Example:
 
Find the adjoint of the matrix.
 
 
 
 
 
 
 
 
 
 
 
 
         
 
Inverse of a Square Matrix
 
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.
 
Note 1: Only a square matrix can have its inverse.
 
Note 2: From the definition, it is clear that if B is the inverse of A, then A is the inverse of B.
 
Note 3: Inverse of A is denoted by A-1, thus B = A-1 and
 
AA-1 = A-1A=I.
 
Theorem:
 
The inverse of a square matrix if it exists, is unique.
 
Let A be an invertible square matrix. If possible, let B and C be two inverse of A.
 
Then AB = BA = I.
 
AC = CA = I (by def. of inverse)
 
Now,
 
B = BI = B(AC)
 
= (BA)C [ Matrix multiplication is associative]
 
= IC = C
 
i.e., B = C
 
Hence the inverse of A is unique.
 
Theorem
 
If A and B are two invertible matrices of the same order, then (AB)-1 = B-1A-1.
 
Proof:
 
From the definition of inverse of a matrix, we have
 
(AB)(AB)-1 = I
 
or A-1 (AB)(AB)-1 = A-1 I (Pre-multiplying both sides by A-1)
 
or (A-1A) B (AB)-1 = A-1 (Since A-1 I = A-1)
 
or I B (AB)-1 = A-1
 
or B (AB)-1 = A-1
 
or (B-1B)(AB)-1 =B-1A-1
 
or I(AB)-1= B-1A-1
 
or (AB)-1 = B-1A-1
 
Theorem
 
 
 
 
 
Elementary Transformation
 
Elementary transformations are of the following three types:
 
Interchange of any two rows (or columns)
 
The multiplication of the elements of a row (or column) by a non-zero number.
 
The addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by any number.
 
Any elementary operation is called a row transformation or a column transformation according as it applies to rows or columns.
 
Definition
 
Let Ri denotes the ith row of the matrix A = [aij] then the elementary row operations on the matrix A are defined as:
 
 
 
3. Ri g Ri + kRj means multiply each element of jth row by k and add it to the corresponding elements of ith row.
 
The corresponding column transformations are
 
 
Properties of adjoint of a matrix
 
1. A.(adj A) = (adj A). A = |A| I
 
2. adj (AB) = (adj B) . (adj A).
 
Non-singular Matrix
 
A square matrix A is said to be non-singular if its determinant value is non-zero.
 
i.e.,
 
 
Singular Matrix
 
A square matrix A is said to be singular if |A| = 0.
 
Properties of Inverse of Matrix
 
 
In other words, a square matrix A is invertible if and only if A is a non-singular matrix.
 
 
(c) If A and B are invertible square matrices, then
 
(AB)-1 = B-1 A-1
 
 
(d) If A and B are two non-singular square matrices of the same order, then AB and BA are also non-singular matrices of the same order.
 
Consistency and Inconsistency of a System of Linear Equations
 
A system of linear equations is said to be consistent if it has a solution. This means that the solution satisfies all the equations in the system simultaneously.
 
If a system of linear equations has no solution, then it is said to be inconsistent.
 
  
     
   
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