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Unit Two
B.Tech Ist Semester
Content
Elementary Transformations (Operation)
Rank of The matrix
Normal Form
Solution of System of Linear Eqn.
Vectors
Linear dependence & independence
Linear &Orthogonal transformation
Content
Eigen value, Eigen vectors and there properties
Cayley-Hamilton Theorem
Diagonal Form of Matrix
Similar Matrix
Quadratic form of Matrix
Matrix A matrix is a rectangular array of numbers. We
subscript entries to tell their location in the array
aij " i th row"
" j th column"
rows m× n
row a11 a12 a13 K a1n
a a22 a23
K a2 n
21
A = a31 a32 a33 K a3n
Matrices are
M M M identified by
their size.
am1 am 2 am 3
K amn
Matrix
1× 5 [3 − 1 5 0 2]
Row Matrix
4× 1 4× 4
− 2 2 − 1 − 2 4
6 −1 3
Column Matrix
5 7
1 − 2 5 − 8 9
4 7 9 0
− 3 Square Matrix
Matrix
A matrix that has the same number of rows as
columns is called a square matrix.
a11 a12 a13 a14
a a22 a23
a24
A= 21
a31 a32 a33 a34
a41 a42 a43 a44
Matrix
Diagonal Matrix Unit Matrix
a11 0 0 ............0 1 0 0 ............0
0 a 0 ............0 0 1 0 ............0
22
0 0 a33 ...........0 0 0 1............0
........................... ...........................
........................... ...........................
0 0 0 0 ann n×n
0 0 0 ........ 0 1 n×n
0 0 0 ............0
0 0 0 ............0
0 0 0............0
Null Matrix
...........................
...........................
0 0 0 ........ 0 0 n×n
Determinant
A determinant is a real number associated with every
square matrix. The determinant of a square matrix A is
denoted by "det A" or | A |.
a b
The determinant of 2x2 matrix is = ad − bc
c d
and 3x3 matrix is
a1 a2 a3
b2 b3 b3 b1 b1 b2
b1 b2 b3 = a1 − a2 + a3
c 2 c2 c3 c1 c1 c2
c1 c2 c3
e.g:
1 2 1
− 1 3 4
(a) Find the determinant of the matrix .
5 2 1
a a2 a3 −1
(b) If b b2 b 3 − 1 = 0 in which a,b,c are different,
c c2 c3 − 1
show that abc = 1 .
Properties of the determinant:
i. A determinant of a matrix is unaffected by changing
its rows(columns) into columns (rows).
ii. If two rows(columns) of a determinant are
interchange then the numerical value of determinant
remains same but change its sign.
iii. If two rows(columns) of a determinant are identical then the
value of determinant is zero.
iv. If each element of any row(column) are multiplied by the same
factor ,then whole determinant multiplied by same factor.
v. If each element of a column consist m terms ,the determinant
can be expressed as the sum of m determinants.
Elementary Transformations
Operations that can be performed without
altering the solution set of a linear system
1. Interchange any two rows
2. Multiply every element in a row by a nonzero constant
3. Add elements of one row to corresponding
elements of another row
Some Result of E-operation
i. Any E-row operation on the product of two matrices
is equivalent to the same operation on the pre-factor.
i.e. for E-row operation R, R(AB)=R(A).B .
ii. Any E-column operation on the product of two
matrices is equivalent to the same operation on the
post-factor.
i.e. for E-column operation C, C(AB)=A.C(B).
Inverse of a Matrix
If A is a square matrix s.t. |A|≠0, then a matrix B is
called the inverse of the matrix A if
AB=BA=I (identity matrix)
It is denoted by A-1 .
Inverse of a matrix by E-operation
(Gauss-Jordan method):
Let A be a square matrix s.t. |A|≠0, then A=IA
Appling E-row operation on A=IA, to reduce L.H.S.
matrix A in to Identity matrix and in R.H.S. Identity
matrix I reduced in a new matrix B(say).i.e.
R(A)=R(I).A reduced to I=BA…………………(i)
Post multiplying (i) by A-1 , we have
A-1 =B
e.g. :
Use Gauss-Jordan method to find the inverse of the
2 0 − 1
matrix 5 1 0 .
0 1 3
Echelon Form
All nonzero rows (rows with at least one nonzero
element) are above any rows of all zeroes, and
The leading coefficient (the first nonzero number
from the left, also called the pivot) of a nonzero row is
1 and always strictly to the right of the leading
coefficient of the row above it.
All entries in the column below a leading 1 are zero.
1 # # #
0 1 #
#
0
0 1 #
e.g.: To change the Matrix A in echelon form by elementary
transformation.
5 3 14 4
0 1 2 1
A=
1 − 1 2 0
Minor of Matrix
If A be mxn matrix then determinant of sub matrix of a
matrix A is called the Minor of the matrix.
e.g.: Find all the minors of the matrix
2 1 − 3 − 6
3 − 3 1 2
1 1
1 2 .
Rank of the Matrix
The order of the highest non-zero minor of a mxn
matrix is called Rank of the matrix and is denoted by
ρ(A).
Note: Number of non-zero row in the echelon form of
the matrix is also the rank of matrix.
e.g.:
(a) Determine the rank of the following matrix:
0 1 −3 − 1
1 0 1 1
3 1 0 1
1 1 −2 0
(b) Reduce the following matrix in Normal Form:
2 1 − 3 − 6
3 − 3 1 2
1 1
1 2
Normal form of a Matrix
If A is an mxn matrix and by a series of E-operation, it
can be find in the following form
I r 0 I r
0 , 0 , [I r 0], [I r ] ,where Ir is the unit
0
matrix of order r.
Note:
For mxn matrix A of rank r, to find square matrices P &
Q of order m and n respectively, such that PAQ is in
normal form.
P and Q can be formed by applying E-operation on
A=IAI.
1 1 2
e.g.: For the matrix A = 1 2 3
find the non-
0 − 1 − 1
singular matrix P and Q such that PAQ is in normal
form.
Solution of a system of linear
Equation
a1 x + b1 y + c1 x = d1
Consider of a system of linear eqn . a2 x + b2 y + c2 x = d 2
In matrix notation we have a3 x + b3 y + c3 x = d 3
AX = B …………..(i)
a1 b1 c1
Where A = a2 b2 c2 is called the co-efficient
matrix. a3 b3 c3
d1
x
d is determine
X = y is variable matrix and B = 2
matrix.
z d 2
If B≠0 then eqn. (i) is called non-homogeneous system
of equation, if B=0 then (i) is homogeneous system of
equation.
Solution of non-homogeneous system of linear eqn.
For a system of non-homogeneous linear equation AX=B
i. If ρ[A:B]≠ρ(A), the eqn. have no solution, system is
called inconsistent.
ii. If ρ[A:B]=ρ(A)= number of variable, the system has a
unique solution.
iii. If ρ[A:B]=ρ(A)< number of variables, the system has
an infinite no. of solutions.
Solution of homogeneous system of linear eqn.
For a system of homogeneous linear equations AX=0
X=0 is always a solution, is called trivial solution.
Thus a homogeneous system is always consistent.
A system of homogeneous linear equation has either
the trivial solution or infinite number of solutions.
(i) if ρ(A)= number of variable, the system has only
trivial solution.
(ii) if ρ(A)< number of variable, the system has an
infinite number of non-trivial solution.
e.g:
a) Test the consistency of the equation x+2y-z=3,3x-
y+2z=1, 2x-3y+7z=5, 2x-2y+3z=2, x-y+z=-1, and
solve them.
b) For what values of λ the equation 3x-y+4z=3, x+2y-
3z=-2, 6x+5y+ λz=-3 have
(i) no solution.
(ii) a unique solution
(iii) an infinite number of solution.
Vector
Any ordered n-tuple of numbers is called an n-vector.
If x1 , x2 , ................xn be any n numbers then the ordred
n-tuple X = ( x1 , x2 ,...............xn ) in called n-vector.
e.g.: In 3D the coordinates are represent with 3-
vectors (x,y,z).
Linearly dependent and independent of
vector.:
A set of r vectors X1, X2, …………., Xr is said to be linearly
dependent if there exist r scalars k1, k2, ………., kr not all
zero , such that
k1 X 1 + k 2 X 2 + ......... + k r X r = 0
and X1, X2, …………., Xr is said to be linearly
independent, ∀ k1 X 1 + k 2 X 2 + ......... + k r X r = 0
⇒ k1 = k 2 = ......... = k r = 0
Note: Let X1, X2, …………., Xr are linearly dependent
vectors, such that, for r scalars k1, k2, ………., kr not all
zero k1 X 1 + k 2 X 2 + ......... + k r X r = 0
if ki ≠ 0 , then
ki X i = − k1 X 1 − k 2 X 2 − ..... − ki −1 X i −1 − ki +1 X i +1....... − k r X r
k1 k2 ki −1 ki +1 kr
⇒ X i = − X 1 − X 2 − ........ − X i −1 − X i +1...... X r
ki ki ki ki ki
then Xi is a linear combination of the remaining
vectors k1, k2, ……… ki-1, ki+1 ,………..kr .
Linear Transformation
If T : R n → R m is a linear transformation, then there
exist a unique mxn matrix A, s.t.
T(X)=AX for all X ∈ Rn .
Let P(x,y) in a plane transformation, transform to the
point P’(x’,y’) under the reflection in co-ordinate axes,
then the co-ordinate of P’ can be expressed in terms of
those of P by the linear transformation of the form
x' = a1 x + b1 y
or in matrix notation x' = a1 b1 x
y ' a b y
y ' = a2 x + b2 y
or X’=AX . 2 2
Orthogonal Transform
The linear transformation Y=AX, where
y1 a11 a12 .......... a1n x1
y a x
: , A = 21 a22 ..........
2 a2 n , X = :2
Y=
: ...... ....... ........... ..... :
yn
am1 am 2 ........... amn xn
is said to be orthogonal if it transformation
2 2 2 2 2 2
y1 + y2 + ........... + yn in to x1 + x2 + .............. + xn .
Note : If the transformation matrix is non-singular , i.e.
if |A|≠0, then the linear transformation is called non-
singular or regular transformation, if A is singular, i.e.
|A|=0 then is called singular.
e.g.: Show that the transformations
y1 = x1 − x2 + x3
y2 = 3 x1 − x2 + 2 x3
y3 = 2 x1 − 2 x2 + 3 x3
is non-singular. Find the inverse transformation.
Eigen value
Consider a linear transformation Y=AX. If Y be a matrix
of some scalar multiple(say λ) of X, i.e. Y= λX ,
then AX = λX ⇒ AX − λIX = 0
⇒ ( A − λI )X = 0...................................(i )
eqn. (i) is system of homogeneous linear eqn. These
system will have non-trivial solution if the coefficient
matrix is singular, i.e. | A − λI |= 0...............(ii )
eqn. (ii) are called Characteristic Equation. and
having n roots ,say λ1, λ2,……….., λn which are called
Eigen value or Characteristic Roots or Latent
Roots.
Eigen vectors
For every value of λ, the homogeneous system of eqn.(i)
have non-zero solution
a1
a
2
.
X = which is called an Eigen vector corresponding
.
. to that value of λ.
an
e.g.: Find the Eigen value and Eigen vector of the
matrix.
− 2 2 − 3
2 1 − 6
(i)
−1 − 2 0
8 −6 2
(ii) − 6 7 − 4
2 −1 3
Properties of Eigen Value.
(i) The eigen value of square matrix A and its transpose A’
are the same.
(ii) The sum of the eigen values of a matrix is the sum of the
elements on principal diagonals.
(iii) The product of the eigen values of a matrix A is equal to
|A|. 1
(iv) If λ is an eigen value of a non-singular matrix A, then is
an eigen value of of A-1 . λ
(v) If λ is an eigen value of an orthogonal matrix A, then is 1
an eigen value of of A-1 . λ
(vi) If λ1 , λ2 ,........, λn are the eigenmvalues of a matrix A, then
A m has the eigen values λ m , λ ,........, λ. m
1 2 n where m∈ Z +
Cayley Hamilton Theorem
Every square matrix satisfies it’s characteristic equation.
i.e. if the characteristic equation of the nth order matrix A
is , | A − λI |= (−1) n λn + a1λn −1 + a2 λn − 2 + ............. + an = 0
then (−1) n An + a1 An −1 + a2 An −2 + ............. + an I = 0
Note: According to Cayley Hamilton theorem
(−1) n An + a1 An −1 + a2 An − 2 + ............. + an I = 0
Multiplying this by A-1 we have
(−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I + an A−1 = 0
1
⇒ A = − [(−1) n An −1 + a1 An − 2 + a2 An −3 + ............. + an −1 I ]
−1
an
e.g.: (a)Verify Cayley Hamilton theorem for the matrix A
and find A-1 7 2 − 2
where A = − 6 − 1 2 .
6
2 − 1
1 4
(b) Find the characteristic equation of the matrix A =
2 3
and use it to find the matrix represented by
A5 + 5 A4 − 6 A3 + 2 A2 − 4 A + 7 I
Also express A5 − 4 A4 − 7 A3 + 11A2 − A − 10 I as a linear
polynomial in A.
Diagonalization of Matrix
If a square matrix A of order n has n linearly independent
eigen vectors then there exist a matrix B such that B-
1AB is a diagonal matrix.
8 −6 2
A = − 6 7 − 4
e.g.: Diagonalise the matrix
2 −4 3
Similar Matrix
Let A and B be square matrices of the same order. The
matrix A is said to be similar to the matrix B if there
exist an invertible square matrix P, s.t. A=P-1BP.
e.g.: Show that the matrices
5 5 1 2
A= and B = are similar.
− 2 0 − 3 4
Quadratic Form
A real quadratic form is an homogeneous expression of
the form n n
Q = ∑∑ aij xi x j in which the total
i =1 j =1
power in each term is 2.
For instance, if a h f x
A = h b g , X = y and X ' = [x y z ]
f
g c
z
then X ' AX = ax 2 + by 2 + cz 2 + 2hxy + 2 gxz + 2 fyz
which is quadratic form.
Canonical Form
n n
If the quadratic form X ' AX = ∑∑ aij xi x j can be reduced to
i =1 j =1
n
the quadratic form Y ' BY = ∑ λi yi 2 by a non-singular linear
i =1
transformation X=PY then Y’BY is called the canonical
form of given one.
If B = P' AP = diag (λ1 , λ2 ,............, λn ) , then
n
X ' AX = Y ' BY = ∑ λi yi
2
i =1
. Where B=P’AP.
Index and Signature of the
Quadratic form
The number of positive terms in the canonical form is
called the index of the quadratic form.
Let p be the number of positive terms and rank of
matrix is r, then the signature of matrix =2p-r.
e.g.: Reduce3x 2 + 5 y 2 + 3z 2 − 2 xy − 2 yz + 2 zx quadratic
forms to canonical forms by orthogonal
transformation. Write also rank, index and signature.
Nature of Quadratic form
A real quadratic form X’AX in n variables is said to be
i. positive definite if all the eigen values of A is >0.
ii. negative definite if all the eigen values of A is <0.
iii. positive semi-definite if all the eigen values of A is
≥0 and at least one eigen value =0.
iv. negative semi-definite if all the eigen values of A is
≤0 and at least one eigen value =0.
