Chapter-1
Mathematics
CHAPTER 1 Linear Algebra Linear algebra comprises of the theory and applications of linear system of equations, linear transformations and Eigen-value problems.
Matrix Definition A system of “m n” numbers arranged along m rows and n columns Conventionally, single capital letter is used to denote matrices Thus,
A=[
a
a a
a a
a
a
a a a
a a a a
]
ith row, jth column
Types of Matrices 1. Row and Column matrices Row Matrices [ 2, 7, 8, 9]
Column Matrices
[
]
single row ( or row vector) single column (or column vector)
2. Square matrix Same number of rows and columns. Order of Square matrix no. of rows or columns e.g. A = [
] ; order of this matrix is 3
Principal Diagonal (or main diagonal or leading diagonal) The diagonal of a square matrix (from the top left to the bottom right) is called as principal diagonal. Trace of the Matrix The sum of the diagonal elements of a square matrix. - tr (λ A) = λ tr(A) , λ is scalar- tr ( A+B) = tr (A) + tr (B) - tr (AB) = tr (BA)
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Chapter-1
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3. Rectangular Matrix Number of rows Number of columns 4. Diagonal Matrix A Square matrix in which all the elements except those in leading diagonal are zero. e.g. [
]
5. Scalar Matrix A Diagonal matrix in which all the leading diagonal elements are same. e.g [
]
6. Unit Matrix (or Identity Matrix) A Diagonal matrix in which all the leading diagonal elements are ‘ ’. e.g.
=
[
]
7. Null Matrix (or Zero Matrix) A matrix is said to be Null Matrix if all the elements are zero. e.g.
0
1
8. Symmetric and Skew symmetric matrices * Symmetric, when a = +a for all i and j. In other words =A Note :- Diagonal elements can be anything. * Skew symmetric, when a = - a In other words = -A Note :- All the diagonal elements must be zero. Symmetric Skew symmetric a h g h g f] [h b f ] [h g f c g f
Symmetric Matrix
𝐀𝐓 = A
Skew Symmetric Matrix
𝐀𝐓 = - A
9. Triangular matrix A matrix is said to be “upper triangular” if all the elements below its principal diagonal are zeros. A matrix is said to be “lower triangular” if all the elements above its principal diagonal are zeros. a a h g [ [g b ] b f] f h c c Upper triangular matrix Lower triangular matrix 10.
Orthogonal matrix: If A. A = , then matrix A is said to be Orthogonal matrix. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter-1
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11.
Singular matrix: If |A| = 0, then A is called a singular matrix.
12.
Unitary matrix ̅ ) = transpose of a conjugate of matrix A If we define, A = (A Then the matrix is unitary if A . A = For example A=[
13.
],A =[
]
A. A = , and Hence it’s a Unitary Matrix
Hermitian matrix It is a square matrix with complex entries which is equal to its own conjugate transpose. A = A or a = a̅ i For example: 0 1 i Note: In Hermitian matrix , diagonal elements
14.
always real
Skew Hermitian matrix : It is a square matrix with complex entries which is equal to the negative of conjugate transpose. A = A or a = a̅̅̅ i For example = 0 1 i Note: In Skew-Hermitian matrix , diagonal elements
either zero or Pure Imaginary
15.
Idempotent Matrix : If A = A, then the matrix A is called idempotent matrix.
16.
Nilpotent Matrix : If A = 0 (null matrix), then A is called Nilpotent matrix (where K is a +ve integer).
17.
Periodic Matrix : If A
= A (where k is a +ve integer), then A is called Periodic matrix.
If k =1 , then it is an idempotent matrix. 18.
Proper Matrix : If |A| = 1, matrix A is called Proper Matrix.
Equality of matrices Two matrices can be equal if they are of (a) Same order (b) Each corresponding element in both the matrices are equal. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter-1
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Addition and Subtraction of matrices [
a c
b ] d
[
a c
b ] = d
[
a c
a c
b d
b ] d
Rules 1. Matrices of same order can be added 2. Addition is commutative A+B = B+A 3. Addition is associative (A+B) +C = A+ (B+C) = B + (C+A) Multiplication of matrix by a Scalar Every element of the matrix gets multiplied by that scalar.
Multiplication of matrices Condition: Two matrices can be multiplied only when number of columns of the first matrix is equal to the number of rows of the second matrix. Multiplication of (m n) and (n m n p) matrices results in matrix of (m p)dimension 0 n p = m p1. To find Rank always remember to make matrix a triangular matrix (Upper triangular) or (lower triangular) [
] or
[
]
Try to make I, zero then II and then III in any Matrix to find Rank of Matrix, for easy execution of question to find rank.
Determinant An n order determinant is an expression associated with n
n square matrix.
If A = [a ] , Element a with ith row, jth column. For n = 2 ,
a D = det A = |a
a a | = (a
a
-a
a )
Determinant of “order n”
D = |A| = det A = ||
a a
a
a
a
a
a a
| |
a
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Chapter-1
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Minors & Co-factors
The minor of an element in a determinant is the determinant obtained by deleting the row and the column which intersect that element. a a a b b | D=| b c c c Minor of a = |
b c
b | c
Cofactor is the minor with “proper sign”. The sign is given by (-1) belongs to ith row, jth column). A2 = Cofactor of
= (-1)
b | c
(where the element
b | c A
A
A
C
C
C
Cofactor matrix can be formed as |
|
In general
= =0 *
, ,
if i
=j j ,
,
.+
Drill Problem Can the determinant be expanded about the diagonal?
Properties of Determinants 1.
A determinant remains unaltered by changing its rows into columns and columns into rows. a a a a b c b b | b c | = | b | a c c c a b c
2.
If two parallel lines of a determinant are inter-changed, the determinant retains it numerical values but changes in sign. (In a general manner, a row or column is referred as line). a b c a c b c a b b c |= c b | =| c a b | | a | a a b c a c b c a b
3. 4.
Determinant vanishes if two parallel lines are identical. If each element of a line be multiplied by the same factor, the whole determinant is multiplied by that factor. [Note the difference with matrix]. a b c a b c a b c a b c | | =| | a b c a b c THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter-1
5.
If each element of a line consists of sum of the m determinants. a b c d e a b c d e | =| a | a a b c d e a
6.
Mathematics
the m terms, then determinant can be expressed as b b b
c a c |+| a c a
b b b
d d | d
a | a a
b b b
e e | e
8.
If each element of a line be added equi-multiple of the corresponding elements of one or more parallel lines, determinant is unaffected. e.g. By the operation, + p +q , determinant is unaffected. Determinant of an upper triangular/ lower triangular/diagonal/scalar matrix is equal to the product of the leading diagonal elements of the matrix. If A & B are square matrix of the same order, then |AB|=|BA|=|A||B|.
9.
If A is non singular matrix, then |A |=| | (as a result of previous).
10. 11. 12. 13.
Determinant of a skew symmetric matrix (i.e. A =-A) of odd order is zero. If A is a unitary matrix or orthogonal matrix (i.e. A = A ) then |A|= ±1. If A is a square matrix of order n then |k A| = k |A|. | | = 1 ( is the identity matrix of order n).
7.
Multiplication of determinants
The product of two determinants of same order is itself a determinant of that order. In determinants we multiply row to row (instead of row to column which is done for matrix).
Comparison of Determinants & Matrices
Although looks similar, but actually determinant and matrix is totally different thing and its technically unfair to even compare them. However just for reader’s convenience, following comparative table has been prepared.
Determinant
Matrix
No of rows and columns are always equal
No of rows and column need not be same (square/rectangle)
Scalar multiplication: elements of one line (i.e. one row and column) is multiplied by the constant
Scalar multiplication: all elements of matrix is multiplied by the constant
Can be reduced to one number
Can’t be reduced to one number
Interchanging rows and column has no effect
Interchanging rows and columns changes the meaning all together
Multiplication of 2 determinants is done by multiplying rows of first matrix & rows of second matrix
Multiplication of the 2 matrices is done by multiplying rows of first matrix & column of second matrix
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Chapter-1
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Transpose of Matrix Matrix formed by interchanging rows & columns is called the transpose of a matrix and denoted by A . ] Transpose of A= Trans (A)= A ’ = A = 0
Example: A = [
1
Note
A =
If A & B are symmetric, then AB+BA is symmetric and AB-BA is skew symmetric. If A is symmetric, then An is symmetric (n= , , …….). If A is skew-symmetric, then An is symmetric when n is even and skew symmetric when n is odd.
(A + A ) +
(A - A )
= symmetric matrix + skew-symmetric matrix
Adjoint of a matrix Adjoint of A is defined as the transposed matrix of the cofactors of A. In other words, Adj (A)= Trans (cofactor matrix) a Determinant of the square matrix A = [ a a
b b b
c c ] is c
The matrix formed by the cofactors of the elements in A [ A A
C C ] C
a = |a a
b b b
c c | c
is
Also called as cofactor matrix A
A
A
C
C
C
Then transpose of this matrix is [
]
--- This is Adj (A)
Inverse of a matrix
A
|A| must be non-zero (i.e. A must be non-singular). Inverse of a matrix, if exists, is always unique.
If it is a 2x2 matrix 0
=
| |
Drill Problem: Prove (A )
=
1 , its inverse will be
0
1
A
Proof RHS = ( A ) Pre-multiplying the RHS by AB, (A B) ( A ) = A (B. ) A =I Similarly, Post-multiplying the RHS by AB, ( A ) (A B) = (A A ) B= Hence, AB & A are inverse to each other
B=
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Chapter-1
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Important Points 1.
A = A = A, (Here A is square matrix of the same order as that of
2.
0 A = A 0 = 0, (0 is null matrix)
3.
If AB = 0, then it is not necessarily that A or B is null matrix.
)
Also at doesn’t mean A = Example AB = 0 4.
1 .0
1=0
If the product of two non-zero square matrix A & B is a zero matrix, then A & B are singular matrix.
5.
If A is non-singular matrix and A.B=0, then B is null matrix.
6.
AB
7.
A(BC) = (A B)C
8.
A(B+C) = AB AC
9.
AC = AD , doesn’t imply C = D ,even when A
BA (in general)
commutative property is not applicable
Associative property holds Distributive property holds -.
10. If A, C, D be nxn matrix, then If r(A)=n and AC=AD, then C=D. (Prove it. Hint: pre-multiply both sides by A ) 11. (A+B)T = A + 12. (AB)T =
. A
13. (AB)-1 = 14. A A
. A
=A A=
15. (kA)T = k.A (k is scalar, A is vector) 16. (kA)-1 = k 17. (A )
. A
(k is scalar , A is vector)
= (A )
̅ ) (Conjugate of a transpose of matrix = Transpose of conjugate of matrix) 18. (̅̅̅̅ A ) = (A 19. If a non-singular matrix A is symmetric, then A 20. f A is a orthogonal matrix , then A and A
is also symmetric.
are also orthogonal.
21. If A is a square matrix of order n then (i) |adj A|=|A| (ii) |adj (adj A)|=|A|(
)
(iii) adj (adj A) =|A|
A
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Example Demonstrate by example that AB
BA
Solution Suppose, A = 0
1, B= 0
AB = 0
1
1 , BA = 0
1
Example Demonstrate by example that AB = 0
A = 0 or B = 0 or BA = 0
Solution AB = 0
1.0
BA = 0
1 =0
1
1
Example Demonstrate that AC = AD
C = D (even when A
0)
Solution AC = 0
1 .0
1 =0
1
AD = 0
1 .0
1 =0
1
Although AC = AD, but C
D
Example Write the following matrix A as a sum of symmetric and skew symmetric matrix A=[
]
Solution Symmetric matrix =
=
[
(A +A ) =
]= [
Skew symmetric matrix =
{[
]
[
]}
]
(A -A ) = [
]
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Example Check, if the following matrix A is orthogonal. A=
[
]
Solution A =A = [
A.A =A.A =
]
] = [
[
]
= ,
Hence A is orthogonal matrix Elementary transformation of matrix 1. 2. 3.
Interchange of any 2 lines Multiplication of a line by a constant (e.g. k ) Addition of constant multiplication of any line to the another line (e. g.
+p
)
Note
Elementary transformations don’t change the rank of the matrix. However it changes the Eigen value of the matrix. We call a linear system S1 “ ow Equivalent” to linear system S2, if S1 can be obtained from S2 by finite number of elementary row operations.
Gauss-Jordan method of finding Inverse Elementary row transformations which reduces a given square matrix A to the unit matrix, when applied to unit matrix , gives the inverse of A Example Find the inverse of
[
]
Solution Write in the form, Operate
[ +
[
] ,
]
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Chapter-1
Operate
Mathematics
[
] ,
/2 ,
/5 [
,
]
+ [
]
+ [
]
Rank of matrix If we select any r rows and r columns from any matrix A, deleting all other rows and columns, then the determinant formed by these r r elements is called minor of A of order r. Definition: A matrix is said to be of rank r when, i) ii)
It has at least one non-zero minor of order r. Every minor of order higher than r vanishes.
Other definition: The rank is also defined as maximum number of linearly independent row vectors. Special case: Rank of Square matrix Rank = Number of non-zero row in upper triangular matrix using elementary transformation.
Note 1. 2. 3. 4. 5. 6. 7. 8.
r(A.B) min { r(A), r (B)} r(A+B) r(A) + r (B) r(A-B) r(A) - r (B) The rank of a diagonal matrix is simply the number of non-zero elements in principal diagonal. A system of homogeneous equations such that the number of unknown variable exceeds the number of equations, necessarily has non-zero solutions. If A is a non-singular matrix, then all the row/column vectors are independent. If A is a singular matrix, then vectors of A are linearly dependent. r(A)=0 iff (if and only if) A is a null matrix. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Example Find rank of
[
]
Solution ,
,
[
]
,
[
, ]
,
-
[
]
[
]
Hence Rank = 3 (Number of Non zero rows) Example
The rank of a diagonal matrix [ (A) 1
(B) 2
] (C) 3
(D) 4
Solution Option (C) is correct as the number of non-zero elements in the diagonal matrix gives the rank. Example Rank of matrix [ (A) 3
] is (B) 2
(C) 0
(D) 1
Solution Option (D) is correct By doing
, we get [
]
Number of non-zero row is 1. Hence rank is 3.
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Example The rank of [
] is 2 , value of µ =?
(A) 3
(B) 2
(C) 1
(D) 0
Solution Option (C) is correct (Hint: Just by equating the determinant to zero)
Vector space It is a set V of non-empty vectors such that with any two vectors “a and b” in V, all their linear combinations ( , are real numbers) are elements of V. Dimension The maximum number of linearly independent vectors in V is called the dimension of V (and denoted as dim V). Basis A linearly independent set in V consisting of a maximum possible number of vectors in V is called a basis for V. Thus the number of vectors of a basis for V is equal to dim V. Span The set of all linear combinations of given vector ( ) , ( ), …………………… ( ) with same number of components is called the span of these vectors. Obviously, a span is a vector space.
Solution of linear System of Equation For the following system of equations a
x +a
x + - -
-
a
x =k
a
x +a
x + - -
-
a
x =k
-
-
-
-
x + - -
-
a
x =k
a
x +a
-
In matrix form, it can be written as
AX=B
Where, THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter-1
a a
a
x x
a a
=
, [a
Mathematics
a
a
=
]
k k ,
[x ]
= [k ]
A= Coefficient Matrix C = (A, B) = Augmented Matrix r = rank (A), r = rank (C), n = Number of unknown variables (x , x , - - - x ) Meaning of consistency, inconsistency of Linear Equation Consistent mean:
one or more solution (i.e. unique or infinite solution)
Consistent a) Unique solution x+2y = 4 3x +2y = 2 b) Infinite solution
Overlap
x+2y = 3x +6y = 12
Inconsistent
No solution
4 2
x+2y = 4 x +2y = 8
Parallel
4
8
Consistency of a system of equations For non-homogenous equations (A X = B) i) ii) iii)
If r r , the equations are inconsistent i.e. there is no solution. If r = r = n, the equations are consistent and there is a unique solution. If r = r < n, the equations are consistent and there are infinite number of solutions.
For homogenous equations (A X = 0) i) ii)
If r =n, the equations have only a trivial zero solution (i.e. x = x = - - - x = 0). If r
