GATE/ESE
Engineering Mathematics Short
Notes for GATE&ESE
Recommended for
CE/ME/EE/ECE/
CS & IT/CH/PI/XE
Compiled By K. UMAMAHESWARA RAO
, Contents:
Chapter No Name Page No
1 Linear Algebra 2
2 Calculus 8
3 Vector Calculus 13
4 Differential Equations 15
5 Numerical Methods 19
6 Partial Differential Equations 21
7 Complex Variable 23
8 Laplace Transforms 25
9 Fourier Series 26
10 Probability & Statistics 27
1|Page
, Linear Algebra
Square matrix: (v) In a square matrix if each element of a
row / column is zero then the value of its
(i) Symmetric Matrix: AT = A determinant is zero.
(ii) Skew symmetric matrix: A = − A
T 6 9 8
6 3 2
(iii) Orthogonal matrix: AAT = AT A = I
0 0 0
C1 ⊥ C2 , C2 ⊥ C3 , C3 ⊥ C1 (vi) In a square matrix two rows / columns
are identical /proportional then the value of
OR
its determinant is zero.
R ⊥ R , R ⊥ R , R ⊥ R
1 2 2 3 3 1
2 3 5
In orthogonal matrix, All the vectors are 6 9 8
orthonormal
6 9 8
* Every square matrix can express as the sum
of symmetric and skew symmetric matrix. (vii) In any two row or column are interchanged,
then magnitude of det remains same but sign
A + AT A − AT changes.
A = +
2 2 (vii) The determinant value of skew
Symmetric Skew− Symmetric
symmetric matrix of odd order is always
zero.
* Properties of determinant:
(viii) The determinant value of an
(i) A = A
T
orthogonal matrix is always either 1 or -1.
(ix) If A is a square matrix of order ‘n’ and k
(ii) AB = A B
is any scalar then kA = k A
n
(iii) A + B A + B
(x) If A is a non-singular matrix of order ‘n’,
(iv) The det value of a triangular diagonal then A 0 .
matrix is the product of its leading diagonal
elements. (a) A ( adj A ) = A I
2 3 5
(b) 𝐴−1 =
𝐴𝑑𝑗𝐴
0 4 6 |𝐴|
0 0 8 n −1
(c) Adj A = A
3|Page
Engineering Mathematics Short
Notes for GATE&ESE
Recommended for
CE/ME/EE/ECE/
CS & IT/CH/PI/XE
Compiled By K. UMAMAHESWARA RAO
, Contents:
Chapter No Name Page No
1 Linear Algebra 2
2 Calculus 8
3 Vector Calculus 13
4 Differential Equations 15
5 Numerical Methods 19
6 Partial Differential Equations 21
7 Complex Variable 23
8 Laplace Transforms 25
9 Fourier Series 26
10 Probability & Statistics 27
1|Page
, Linear Algebra
Square matrix: (v) In a square matrix if each element of a
row / column is zero then the value of its
(i) Symmetric Matrix: AT = A determinant is zero.
(ii) Skew symmetric matrix: A = − A
T 6 9 8
6 3 2
(iii) Orthogonal matrix: AAT = AT A = I
0 0 0
C1 ⊥ C2 , C2 ⊥ C3 , C3 ⊥ C1 (vi) In a square matrix two rows / columns
are identical /proportional then the value of
OR
its determinant is zero.
R ⊥ R , R ⊥ R , R ⊥ R
1 2 2 3 3 1
2 3 5
In orthogonal matrix, All the vectors are 6 9 8
orthonormal
6 9 8
* Every square matrix can express as the sum
of symmetric and skew symmetric matrix. (vii) In any two row or column are interchanged,
then magnitude of det remains same but sign
A + AT A − AT changes.
A = +
2 2 (vii) The determinant value of skew
Symmetric Skew− Symmetric
symmetric matrix of odd order is always
zero.
* Properties of determinant:
(viii) The determinant value of an
(i) A = A
T
orthogonal matrix is always either 1 or -1.
(ix) If A is a square matrix of order ‘n’ and k
(ii) AB = A B
is any scalar then kA = k A
n
(iii) A + B A + B
(x) If A is a non-singular matrix of order ‘n’,
(iv) The det value of a triangular diagonal then A 0 .
matrix is the product of its leading diagonal
elements. (a) A ( adj A ) = A I
2 3 5
(b) 𝐴−1 =
𝐴𝑑𝑗𝐴
0 4 6 |𝐴|
0 0 8 n −1
(c) Adj A = A
3|Page
