The inverse of a square matrix obtaining by using elementary row or column operations.
Adjoint of a matrix
Consider A =
Let Aij = Cofactor of aij
Transpose of matrix =
According to the definition, adj A =
Adjoint of a matrix
Let A =
The cofactors of the elements of A are
A11 = d, A12 = -c, A21 = -b, A22 = a
A = =
Theorem 1
If A is any given square matrix of order n, then:
A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n.
Singular matrix:
A square matrix is said to be singular if its determinant is zero.
Non-singular matrix:
A square matrix is said to be non-singular if its determinant is not zero.
Theorem 2
If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
Claim: |AB| = |A||B|
Theorem 3
The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order.
Theorem 4
If A is a square matrix of order n, then |(Adj A)| = |A|n-1.
Inverse of a square matrix:
A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I.
If a square matrix, A, is invertible, then it is a non-singular matrix.
Theorem 5
If A is a non-singular matrix, then A is invertible and A-1 = .
The inverse of a square matrix obtaining by using elementary row or column operations.
Adjoint of a matrix
Consider A =
Let Aij = Cofactor of aij
Transpose of matrix =
According to the definition, adj A =
Adjoint of a matrix
Let A =
The cofactors of the elements of A are
A11 = d, A12 = -c, A21 = -b, A22 = a
A = =
Theorem 1
If A is any given square matrix of order n, then:
A(adj A) = (adj A) A = |A| I, where I is the identity matrix of order n.
Singular matrix:
A square matrix is said to be singular if its determinant is zero.
Non-singular matrix:
A square matrix is said to be non-singular if its determinant is not zero.
Theorem 2
If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
Claim: |AB| = |A||B|
Theorem 3
The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B| , where A and B are square matrices of the same order.
Theorem 4
If A is a square matrix of order n, then |(Adj A)| = |A|n-1.
Inverse of a square matrix:
A square matrix, A, is said to be an invertible matrix if there exists a square matrix, B, such that AB = BA = I.
If a square matrix, A, is invertible, then it is a non-singular matrix.
Theorem 5
If A is a non-singular matrix, then A is invertible and A-1 = .