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SUB: MATHEMATICS
MODULE-1A
MODULE-1A
Linear Algebra: Matrix algebra, system of linear equations, Eigen values and
Eigen vectors.
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Linear algebra deals with theory and application of linear system of equations, linear
transformations and Eigen value problems.
Determinants: The expression |
| s called a determinant of second order, and stands
for “a1d1-c1b1”.
Similarly,
is called determinant of third order.
Minor: The minor of any element in a matrix is the determinant obtained by deleting the row
and column which intersect in that element.
Co-factor: The minor multiplied by proper sign is called co-factor. Sign is (-1)i+j where i is
row number and j is column number.
Determinant is sum of the products of the elements of any row or column by the
corresponding co-factors.
Properties of determinants:
a) Determinant remains same by changing its rows into columns and columns into rows.
b) If two parallel lines of determinant are interchanged, the det becomes –ve of original.
c) If two lines are same then det=0.
d) If each element of line multiplied by „k‟ the det is k[det].
e) If each element of line consists of „m‟ terms, the determinant can be expressed as the
sum of „m‟ determinants.
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f) If to each elements of a line be added equi-multiples of the corresponding elements of
one or more parallel lines, the det remains same.
g) If the elements of a determinant ∆ are functions of x and two parallel lines becomes
identical when x=a, then x-a is a factor of ∆.
Matrix: A system of m, n numbers arranged in m rows and n columns and bounded by the
brackets [] is called an m x n matrix.
Diagonal matrix: A square matrix all of whose elements except those in the leading diagonal
are zero.
Unit matrix: Diagonal matrix having 1‟s as its leading diagonal elements.
Symmetric matrix: If AT=A then A is Symmetric.
Skew symmetric matrix: If AT=-A then A is Skew-Symmetric.
Only matrices of the same order can be added or subtracted.

Matrix addition is commutative i.e A+B=B+A.

Matrix addition or subtraction is associative (A+B)-C=A+(B-C).
Multiplication of matrix by scalar k. k[
]=[
]
Multiplication of matrices: Two matrices can be multiplied only when number of columns
in the first is equal to the number of rows in the second.
Amxn Bnxp = Cmxp.

Matrix multiplication is associative i.e (AB)C=A(BC).

Matrix multiplication is distributive i.e A(B+C)=AB+AC.
Idempotent matrix: If A2= A then A is called idempotent matrix.
In Amxn Bnxp = Cmxp

Number of multiplications: mnp

Number of additions: m(n-1)p.
Transpose of a matrix: Rows in columns and columns into rows i.e (AB)T=BTAT.
Adjoint of a square matrix: The matrix formed by the co-factors of the elements in ∆, and
transposed.
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Inverse of a matrix: A-1=
MODULE-1A
, (AB)-1=B-1A-1.
Rank of a matrix: Largest order of any non-vanishing minor of the matrix.
Elementary transformation of a matrix:
a) The interchange of any two rows or columns.
b) The multiplication of any row or column by a non-zero number.
c) The addition of constant multiple of the elements of any row or columns to the
corresponding elements of any other row or column.
Equivalent matrix: Two matrices A and B are said to be equivalent if one can be obtained
from the other by a sequence of elementary transformations. Two equivalent matrices have
the same order and the same rank.
Elementary matrix: It is that, which is obtained from a unit matrix, by subjecting it to any of
the elementary transformations.

Elementary row transformations of a matrix A can be obtained by pre-multiplying A
by the corresponding elementary matrices.

Elementary column transformations of a matrix A can be obtained by post multiplying
A by the corresponding elementary matrices.
Normal form of a matrix: Every non-zero matrix A of rank r can be reduced by a sequence
of elementary transformations to the form [
] called the normal form of A.
Consistency of linear system of equations:
Find the ranks of the co-efficient matrix A and the augmented matrix k, by reducing A to the
triangular form by elementary row operations. Let rank of A is r and rank of k be r'.
a) If r≠r', then the equations are inconsistent, i.e there is no solution.
b) If r=r'=n, then the equations are consistent and there is a unique solution.
c) If r=r'<n, then the equations are consistent and many solutions.
System of linear homogeneous equations:
Find the rank of matrix A and it is r, n variables and m equations.
1. If r=n, then the equations have only trivial solution, x1=0,x2=0,…..
2. If r<n , then (n-r) linearly independent solutions. (Many solutions)
3. If m<n, then many solutions.
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SUB: MATHEMATICS
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Eigen values: |A-λI | is called the characteristic equation of A. The roots of this equation are
called the eigen values.
The sum of eigen values is equal to sum of diagonal (principle) elements of matrix, which is
called trace of the matrix.
Eigen vector:[ A- λI ] [λ]=0 for each λ value there is a solution X = [x1,x2,…xn] which is
called eigen vector.
Properties of eigen values:
1. Any square matrix A and its transpose AT have the same eigen values.
2. The eigen values of a triangular matrix are just the diagonal elements of the matrix.
3. The eigen values of an idempotent matrix are either zero or unity.
4. The sum of eigen values of matrix is the sum of the elements of the principal
diagonal.
5. The product of the eigen values of matrix A is equal to its determinant.
6. If λ is an eigen value of A then ⁄ is eigen value of A-1.
7. If λ is an eigen value of an orthogonal matrix, then ⁄ is also its eigen value. (A-1=AT
is orthogonal)
8. If λ1,λ2,……..λn are eigen values of A then λ1m , λ2m , λnm are eigen values of Am.
Canonical form of matrix is
Quadratic form:
where
=[
]
Matrix of transformation P: It is having eigen vectors of A spectral matrix of A: Â=P-1AP
is a diagonal matrix having λ1, λ2 and λ3 as elements.
Complex matrices:
Conjugate of matrix: If elements are complex numbers A=[aij+bij] then Ᾱ=[aij-ibij] is called
conjugate matrix.
= ̅.
Hermitian matrix:
Skew Hermitian:
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=
̅.
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Properties of Hermitian and Skew hermitian matrix:
1. The elements of principal diagonal are real in Hermitian matrix.
2. The elements of principal diagonal are either zeros or purely imaginary in Skew
hermitian matrix.
3. Any square matrix A can be written as the sum of a Hermitian and Skew hermitian
matrices.
4. If A is a herimitian matrix, then iA is a Skew hermitian.
5. The eigen values of a Hermitian matrix are real.
6. The eigen values of a skew Hermitian matrix are purely imaginary or zero.
Unitary matrix:
1. ŪT = U-1 is called unitary matrix.
2. Inverse of unitary matrix is unitary.
3. Inverse of orthogonal matrix is orthogonal.
4. Transpose of unitary matrix is unitary.
5. Transpose of orthogonal matrix is orthogonal.
6. Product of two unitary matrices is a unitary matrix.
7. Product of two orthogonal matrices is orthogonal.
8. The eigen value of a unitary matrix has absolute value 1.
9. The eigen value of an orthogonal matrix has absolute value 1.
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