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Area of = 1 2 |
2 [^ 8(^6 ^ 5) + 2(^4 ^ (^ 1)) + 1(^20 ^ 6)]
2 [88^ ^6 ^ 26]
= 28 Sq. units
- Find the area of triangle with vertices (4, 7), (1, 3) and (5, 1). Solution:
Area of ABC =
x 1 x 2 x 3
y 1 y 2 y 3
Area of ABC =
2 [4(3^ ^ 1)^ ^ 7(1^ ^ 5) + 1(1^ ^ 15)]
2 [8 + 28^ ^ 14]
2 [22]
= 11 Sq. units Exercise:
- Find the area of the triangle with vertices (3,1) , (-1,3) , (-3,-2)
- Find the area of the triangle with vertices (3, 4) , (5,7) , (-2,-3)
- Find the area of triangle with vertices A(2,1), B(1, 4) and C( 3, 2).
- Find the area of the triangle whose are ( 1, 5), (3, 1) and (5, 7).
- Find the area of the triangle whose vertices are (3, 1), ( 1, 3) and ( 3, 2).
- Find the area of triangle whose vertices are (2, 3), (5, 7) and ( 3, 4)
- Find the area of triangle ABC where, A ≡ (1, 2), B ≡ ( 6, 1) and C≡ (0, 8).
Matrices-
Significance of Matrices: Matrices can be used to compactly write and work with multiple linear equations, referred as system of linear equations, simultaneously
Definition:-
A set of 𝑚 × 𝑛 numbers arranged in a rectangular form of m rows & n columns enclosed between
a pair of square brackets is called a matrix of order 𝑚 × 𝑛 (read as m by n).
Matrices are generally denoted by capital alphabets & its elements are denoted by small alphabets.
For e.g. 𝐴 = [
]
3×
; 𝐵 = [
]
𝑚×𝑛 In short, 𝐴 = [𝑎𝑖𝑗]𝑚×𝑛 where 𝑖 = 𝑁𝑜. 𝑜𝑓 𝑟𝑜𝑤𝑠 1,2,3, …. , 𝑚 & 𝑗 = 𝑁𝑜. 𝑜𝑓 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 1,2,3, ….. , 𝑛.
Order of a matrix:-
The order of a matrix is defined as 𝑚 × 𝑛 if it contains m rows & n columns. Examples:
- 𝐴 = [2 3 −1]^ Order of A is 1 × 3
- 𝐵 = [
] Order of B is 3 × 2
3. 𝐶 = [^5 6
] Order of C is 2 × 3
- 𝐷 = [
] Order of D is 2 × 1
Types of matrices:-
- Row matrix: Matrix having only one row is called row matrix. For e.g. : 𝐴 = [2 3 −1].
- Column matrix: Matrix having only one column is called column matrix. For e.g. : 𝐷 = [^8 4
].
- Square matrix : Matrix having equal number of rows & columns is called square matrix
For e.g. 𝐴 = [
]
Note: In matrix A, elements 2, 3, 4 are diagonal elements and remaining are non-diagonal elements.
- Diagonal matrix: A square matrix where all non-diagonal elements are zero is called a
diagonal matrix. For e.g. : 𝐷 = [
]
- Scalar matrix: A diagonal matrix where all diagonal elements are equal is called a scalar
matrix. For e.g. : 𝐾 = [
]
= [
]
- If A = (^)
1 ,^ B =^
0 ,^ C =^
2 then find 5A^ ^ 3B + 2C Solution: 5A 3B + 2C
=5 (^) ^
1 ^3
0 + 2^
= ^
5 ^
0 +^
= ^
- Find the value of x and y satisfying the equation
[
𝑦 2 4 ] + [
] = [
]
Solution: [
] + [^3 1
] = [^4 2
]
∴ [
𝑦 + 4 2 + 3 4 − 2] = [
]
∴ [
𝑦 + 4 5 2 ] = [
]
By using equality of matrices, 𝑥 + 1 = 2 and 𝑦 + 4 = 6 ∴ 𝑥 = 1 & 𝑦 = 2
- If A = (^)
3 ,^ B =^
4 , find the matrix ‘X’ such that 2A + X = 3B Solution: 2A + X = 3B X= 3B 2A
X= 3 (^)
4 ^2
12 ^
X =
- If 𝐴 = [
] , 𝐵 = [
] , 𝐶 = [
]
Then prove that (𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶) Solution: 𝐿. 𝐻. 𝑆. = (𝐴 + 𝐵) + 𝐶
= ( [^2
] + [^4 −
] ) + [^1
]
= [^6
] + [^1
]
= [
]
= [^2
] + ( [^4 −
] + [^1
] )
= [
] + [
]
= [
]
Exercise:
- If A = (^)
1 and B =^
2 , find 2A^ ^ 3B
- If X = (^)
4 ,^ Y =^
3 ,^ Z =^
- Show that 3X + Y = Z
- If A = (^) ^
7 ,^ B =^
6 , find 2A + 3B^ ^ 4I
- If A =
, B =
, C =
, find 3A + 4B 2C
- If A = (^)
3 ,^ B =^
2 ,^ C =^
Verify that (A + B) + C = A + (B + C)
- If A =
x
3
2y
, B =
2y + 5
9
and if 3A = B , find x and y
- If A =
, B =
, verify that A + B = B + A
Matrix multiplication:
The product of two matrices A and B is possible only if the number of
columns in A is equal to the number of rows in B.
Let A = [aij] be an m n matrix
B = [bij] be an n p matrix.
Order of A× B is m×p
Method of Multiplication of two matrices:
3𝐼 = 3 [
] = [
]
∴ 𝐴^2 − 3I = [
] − [
]
∴ 𝐴^2 − 3I = [
]
- Find x and y if
4
3 ^2
x y
Solution:
(^4)
3 ^2
x y
12 ^
x y
x y
x y
x y
4 =^
x y x=2 and y =
- Find x, y, z if
x y z Solution:
x y z
[
] × [
] =
x y z
x y z
x y z
x y z ∴ x = 31 y = 53 z = 19
If A showthat A 8 Ais scalarmatrix 4 4 2
Solution : A
A^2 A. A
A^2
A^2 8 A
III. (𝐴 × 𝐵)′^ = 𝐵′^ × 𝐴′
Symmetric Matrix :
Definition : In a matrix A, if 𝑎𝑖𝑗 = 𝑎𝑗𝑖 for all 𝑖 𝑎𝑛𝑑 𝑗 then matrix is known as symmetric matrix
i.e. if 𝐴 = 𝐴′^ then matrix is known as symmetric matrix.
For e.g. 𝐴 = [
]
Skew Symmetric Matrix:
Definition : In a matrix A, if aij = −aji for all 𝑖 𝑎𝑛𝑑 𝑗 then matrix is known as skew symmetric
matrix i.e. if 𝐴 = −𝐴′^ then matrix is skew symmetric matrix.
For e.g. 𝐴 = [
]
𝐼𝑓 𝐴𝐴′^ = 𝐴′𝐴 = 𝐼 then A is called orthogonal matrix.
Solved examples:
- If A = (^)
0 and B =^
0 , verify that (A + B)
T = AT + BT
Solution:
A = (^)
0 B =^
AT^ =
BT^ =
A + B =
0 +^
A + B =
(A + B)T^ =
AT^ + BT=
From (1) and (2) (A + B)T^ = AT^ + BT
- If A =
and B =
1 then verify that^ (AB)^ = BA
Solution:
A = (^)
5 and^ B =^
A =
5 and^ B^ =
AB =
AB =
(AB) =
… (i)
B A =
B A=
… (ii)
From (i) and (ii) (AB) = B A
- If A =
and B =
, verify that (AB) = B A
Solution:
Given A=
and B =
A=
and B =
A square matrix A is called non-singular, if det (A) or |A| 0.
Solved Example:
- If A = (^)
3 , B =
Show that the matrix AB is non-singular.
Solution: Given A = (^)
3 B =
AB =
AB =
1 + 6 + 3 =^
|AB| =
|AB| = 17 0
AB is a non-singular matrix.
Exercise:
- Prove that the matrix (^)
9 is nonsingular matrix.
- If A = (^) ^
4 , B =^
3 Show that AB is non-singular matrix.
- If A = (^) ^
3 B =^
- 2 decide whether AB is singular or non-singular matrix?
Adjoint of a matrix:
Adjoint of a matrix is the transpose of co-factor matrix
∴ Adj A = [cij]
t
Co-factor matrix is a matrix of co-factors=[𝑐𝑖𝑗]
𝑤ℎ𝑒𝑟𝑒 𝑐𝑖𝑗 = (−1)𝑖+𝑗^ × 𝑀𝑖𝑗 where Minor 𝑀𝑖𝑗 = determinant of matrix obtained by deleting ith^ row & jth column of given matrix.
Solved examples:
- If A=[
] , find Adj A
Solution: Given A=[
]
𝑐 11 = (−1)1+1^ × |^4
𝑐 12 = (−1)1+2^ × |^2
𝑐 13 = (−1)1+3^ × |^2
𝑐 21 = (−1)2+1^ × |
𝑐 22 = (−1)2+2^ × |
𝑐 23 = (−1)2+3^ × |
𝑐 31 = (−1)3+1^ × |
𝑐 32 = (−1)3+2^ × |
𝑐 33 = (−1)3+3^ × |−1^1
∴ 𝑀𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑐𝑜𝑓𝑎𝑐𝑡𝑜𝑟𝑠 = 𝐶 = [
]
∴ 𝐴𝑑𝑗 𝐴 = 𝐶𝑡^ = [
]
Exercise:
- Find adjoint of matrix of A. If A =
- Find the adjoint of matrix of A if A =
- Find adjoint of the matrix A. If A =
- If A =
Find adjoint of A
Inverse of a matrix:
If matrix A is a non-singular matrix and if there exists a matrix B such that 𝐴 × 𝐵 = 𝐵 × 𝐴 = 𝐼 then matrix B is the inverse of A. Notation: Inverse of A = 𝐴−
A
1 0
3 6 8
∴ A^ exists
1
To find cofactor matrix
C
C 12
C
C
C 22
C
C
C 32
C
Cofactor matrix
AdjA
AdjA A
A
Exercise:
- Find the inverse of the matrix A =
by using adjoint matrix.
- Find the inverse of the matrix A =
by using adjoint method.
- Find A ^1 by adjoint method if A =
- Find inverse of the matrix A =
using adjoint method.
- Find the inverse of matrix by adjoint method A =
Suppose a 1 x+ b 1 y + c 1 z = d 1 a 2 x+ b 2 y + c 2 z = d 2 a 3 x+ b 3 y + c 3 z = d 3 are the simultaneous equations.
These equations can be represented in matrix form as follows:
[
] [
] = [
] i.e. 𝐴 × 𝑋 = 𝐵 where
𝐴 = [
] ; 𝑋 = [
] ; 𝐵 = [
]
∴ 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑋 = 𝐴−1^ × B where 𝐴−1^ = 1 det 𝐴 × 𝐴𝑑𝑗 𝐴
× [
]
× [
]
[
] = [
]
- Using matrix inversion method, solve the following system of equations
x+ y + z = 3 , 3x - 2y + 3z = 4 5x + 5y +z = 11
Solution: x+ y + z = 3
3x - 2y + 3z = 4 5x + 5y +z = 11
Given system of equation can be written in matrix form
z
y
x
AX B
X A ^1 B
Where
B
z
y
x A X
Here, 5 5 1
A
17 12 25
A 20 0
A ^1 exists
To find the cofactor matrix of A
C
C 12
C
C 21
C 22
C 23
C
C 32
C
cofactormatrix
AdjA
AdjA A
A
∴ 𝑋 = 𝐴−1^ × B
[
] =
1 20 [
] [
]
