Class 11 Chapter3
3.1 Introduction *row matrix and column matrix are also called row
Matrix: vector and column.
A rectangular array of numbers enclosed by a pair Principal Diagonal:
of bracket is called a matrix. The diagonal from upper left corner to the lower
2 −1 3 right corner of a square matrix is called principal or
𝑒. 𝑔; [ ]
−5 4 7 main diagonal or leading diagonal.
Note: e.g.;
i) Matrices are denoted by capital letters 𝑎11 𝑎12 𝑎13
such as A,B,C…,X,Y,Z
[ 21 𝑎22 𝑎23 ]
𝑎
ii) The elements or entries of a matrix as 𝑎31 𝑎32 𝑎33
denoted by small letters such as
𝑎, 𝑏, 𝑐, … , 𝑥, 𝑦, 𝑧. the entries 𝑎11 , 𝑎12 , 𝑎13 form the principal diagonal
iii) The horizontal lines of elements are Secondary diagonal:
called rows of a matrix. The diagonal from lower left corner to the upper
iv) The vertical lines of elements are called right corner of a square matrix is called secondary
columns of a matrix. diagonal or leading diagonal.
Order of a matrix. e.g.;
The number of rows and columns of a matrix is 𝑎11 𝑎12 𝑎13
called order of a matrix. [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33
pk
i.e.; if a matrix has m rows and n column then its
order is 𝑚 × 𝑛 (read as m-by-n). The entries 𝑎13 , 𝑎22 , 𝑎𝑛𝑑 𝑎31 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦
2 −1 3
𝐴=[ ] 𝑖𝑡𝑠 𝑜𝑟𝑑𝑒𝑟 𝑖𝑠 2 × 3 Diagonal.
1 4 7 s.
𝐵 = [1 4 6] 𝑖𝑡𝑠 𝑜𝑟𝑑𝑒𝑟 𝑖𝑠 1 × 3 Diagonal Matrix:
*the matrix A is called real if all of its elements are A square matrix in which all elements except the
te
real. diagonal are zero is called a diagonal matrix.
Types of Matrices 3 0 0
1 0
e.g.; [ ],[0 3 0]
no
Row Matrix: 0 2
A matrix which has only one row i.e; a matrix of 0 0 3
Identity Matrix:
order 1 × 𝑛 is called row matrix. A diagonal matrix whose all elements of the main
e.g.;[𝑎11 𝑎12 𝑎13 ] ,
sy
diagonal are 1 is called identity matrix denoted by
[1 2 3 ] etc. 𝐼𝑛
Column Matrix: 1 0 0
A matrix which has only one column. i.e.; a matrix 1 0
e.g.; [ ] ,[0 1 0]
ea
of order 𝑚 × 1 is called column matrix. e.g.; 0 1 2×2
0 0 1 3×3
𝑎11 1
[𝑎12 ] , [2] 𝑒𝑡𝑐. Scalar Matrix:
𝑎13 3 A diagonal matrix whose all elements of the main
Rectangular Matrix.
diagonal are same is called scalar matrix.
A matrix whose number of rows and columns are 𝑘 0 0
not equal is called rectangular matrix.eg.; 2 0
e.g.; [ ] ,[0 𝑘 0]
2 −1 3 𝑎11 𝑎12 𝑎13 0 2 2×2
[ ] , [𝑎 ] 0 0 𝑘 3×3
1 4 7 @1 𝑎22 𝑎23
Square Matrix: Needs to remember
A matrix whose number of rows and columns are Square Matrix: A matrix having m rows and n
equal is called square matrix. e.g.; columns with m=n is called square matrix.
1 2 𝑎11 𝑎12 Rectangular Matrix: A matrix having m rows and n
[ ] 𝑎𝑛𝑑 [𝑎 ] columns with 𝑚 ≠ 𝑛 is called rectangular matrix.
3 4 21 𝑎22
Null Matrix: Diagonal Matrix: let 𝐴 = [𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
A matrix whose all elements are zero is called null of order n, if 𝑎𝑖𝑗 = 0∀ 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑎𝑖𝑗 ≠
matrix. 0 𝑓𝑜𝑟
0 0 0 0 0 𝑖 = 𝑗 𝑠𝑜𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙
e.g.; [ ], [ ]
0 0 0 0 0 𝑜𝑓 𝐴 𝑚𝑎𝑦 𝑏𝑒 𝑧𝑒𝑟𝑜 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑎𝑙𝑙, 𝑡ℎ𝑒𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 𝑖𝑠
𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥.
1|Page
,Class 11 Chapter 3
Identity Matrix: let 𝐴 = If the product AB is defined, then the order
[𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 of the product can be illustrated as given
𝑜𝑟𝑑𝑒𝑟 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑖𝑗 = 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 = 𝑗 below.
Then A is called unit matrix or identity matrix Order of A 𝑚×𝑛
denoted by 𝐼𝑛 . 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝐵 𝑛×𝑝
Null Matrix: A matrix of order 𝑚 × 𝑛 with all Order of AB 𝑚×𝑛
elements zero is called null matrix. NOTE:
Scalar Matrix: let 𝐴 = Powers of square matrices are defined as 𝐴2 = 𝐴 ×
[𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝐴
Order n. if 𝑎𝑖𝑗 = 0 ∀ 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑖𝑗 = 𝑘(𝑠𝑜𝑚𝑒 𝑛𝑜𝑛 𝐴3 = 𝐴 × 𝐴 × 𝐴 … … 𝑡𝑜 𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑠.
Determinant of 𝟐 × 𝟐 Matrix:
𝑧𝑒𝑟𝑜 𝑠𝑐𝑎𝑙𝑎𝑟) ∀ 𝑖 = 𝑗 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑜𝑚𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑖𝑠
𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑐𝑎𝑙𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥. The determinant of a matrix is denoted by enclosing
Equal matrices: its square array between vertical bars instead of
Two matrices of the same order are said to be equal brackets.
𝑎 𝑐 𝑎 𝑐
if corresponding entries are equal. e.g; 𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
𝑏 𝑑 𝑏 𝑑
Addition of Matrices:
Two matrices can be added if both have same |𝐴| = 𝑎𝑑 − 𝑏𝑐
order. 𝑓𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙
*addition is done by adding corresponding entries 2 −1 2 −1
𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
pk
of the matrices. 4 3 4 3
*Matrices of different orders cannot be added. |𝐴| = (2)(3) − (4)(−1) = 6 + 4 = 10
Transpose of matrix: Hence the determinant of a matrix is the difference
s.
Transpose of a matrix is denoted by 𝐴𝑡 can be of the product of the entries in the two diagonals.
obtained by interchanging rows into column or Singular Matrix:
A square matrix A is said to be singular if |𝐴| = 0
te
column into rows.
Scalar multiplication: Example:
8 4 8 4
𝑙𝑒𝑡 𝐴 = [𝑎𝑖𝑗 ]𝑏𝑒 𝑎𝑛 𝑚 × 𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
no
2 1 2 1
𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑛𝑑 𝑘 𝑏𝑒 𝑎 𝑠𝑐𝑙𝑎𝑟 |𝐴| = (8)(1) − (4)(2) = 8 − 8 = 0
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑘 𝑘 𝑎𝑛𝑑 𝐴 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 Non-Singular Matrix:
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓𝐴 𝑏𝑦 𝑘. A square matrix A is said to be Non- singular if |𝐴| ≠
sy
𝑖. 𝑒 𝑘𝐴 = [𝑘𝑎𝑖𝑗 ] 0
𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝑘𝐴 𝑖𝑠 𝑚 × 𝑛 Example:
*𝑖𝑓 𝑛 𝑖𝑠 𝑎 + 𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 , 1 2 1 2
ea
𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
𝑡ℎ𝑒𝑛 𝐴 + 𝐴 + ⋯ , 𝑡𝑜 𝑛𝑡𝑒𝑟𝑚𝑠 = 𝑛𝐴 4 9 4 9
Subtraction of matrices: |𝐴| = (1)(9) − (2)(4) = 9 − 8
𝑖𝑓 𝐴 = [𝑎𝑖𝑗 ]𝑎𝑛𝑑 𝐵 = [𝑏𝑖𝑗 ] 𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 = 1 ≠ 0 𝑠𝑜 𝐴 𝑖𝑠 𝑛𝑜𝑛 − 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟
Ad joint of 𝟐 × 𝟐 𝑴𝒂𝒕𝒓𝒊𝒙:
𝑚 × 𝑛, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑑𝑒𝑓𝑖𝑛𝑒 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐵 𝑓𝑟𝑜𝑚 𝐴
𝑎 𝑏
As: 𝑡ℎ𝑒 𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑜𝑓 𝑎 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 = [ ] 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑
𝑐 𝑑
𝐴 − 𝐵 = 𝐴 + (−𝐵) = [𝑎𝑖𝑗 ] + [−𝑏𝑖𝑗 ] 𝑑 −𝑏
𝑏𝑦 𝑎𝑑𝑗𝐴 𝑎𝑛𝑑 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑎𝑑𝑗𝐴 = [ ]
𝐴 − 𝐵 = [𝑎𝑖𝑗 − 𝑏𝑖𝑗 ] −𝑐 𝑎
𝑖 = 1,2,3, . . , 𝑚, 𝑗 = 1,2,3, … , 𝑛 Inverse of 𝟐 × 𝟐 𝑴𝒂𝒕𝒓𝒊𝒙
𝐴 Let A be a non-singular square matrix of order 2. If
− 𝑏 𝑖𝑓 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓 𝐵 there exist of matrix B such that
𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓 𝐴. 1 0
𝐴𝐵 = 𝐵𝐴 = 𝐼2 𝑤ℎ𝑒𝑟𝑒 𝐼2 = [ ] 𝑡ℎ𝑒𝑛 𝐵 𝑖𝑠
Multiplication of two Matrix: 0 1
𝑐𝑎𝑙𝑙𝑒𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝐴 𝑎𝑛𝑑 𝑖𝑠 𝑢𝑠𝑢𝑎𝑙𝑙𝑦
Two matrix A and B are said to be conformable for
𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴−1
product AB if the number of columns of A is equal i.e.; 𝐵 = 𝐴−1 𝑡ℎ𝑢𝑠
to the number of rows of B. 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼2
*Matrix multiplication is not commutative 𝑖. 𝑒;. Solution Of Simultaneous Linear Equations By
𝐴𝐵 ≠ 𝐵𝐴 Using Matrices:
𝑙𝑒𝑡 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑠. 𝑏𝑒
𝑎11 𝑥1 + 𝑎12 𝑥2 = 𝑏1
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,Class 11 Chapter 3
𝑎21 𝑥1 + 𝑎22 𝑥2 = 𝑏2 Exercise 3.1
𝑤ℎ𝑒𝑟𝑒 𝑎11 , 𝑎12 , 𝑎21 , 𝑎22 , 𝑏1 , 𝑏2 ∈ 𝑅 𝟐 𝟑 𝟏 𝟕
Given system in matrix form Q1. If 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [ ]
𝟏 𝟓 𝟔 𝟒
𝑎11 𝑎12 𝑥1 𝑏 𝒕𝒉𝒆𝒏 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝒊) 𝟒𝑨 − 𝟑𝑨 = 𝑨
[𝑎 𝑎 ] [𝑥 ] = [ 1 ]
21 22 2 𝑏2 𝒊𝒊) 𝟑𝑩 − 𝟑𝑨 = 𝟑(𝑩 − 𝑨)
𝑎11 𝑎12 𝑥1 𝑏
𝑙𝑒𝑡 𝐴 = [𝑎 ] , 𝑋 = [𝑥 ] , 𝐵 = [ 1 ] Solution:
21 𝑎22 2 𝑏2 4𝐴 − 3𝐴 = 𝐴
𝑖𝑓|𝐴| ≠ 0 𝑡ℎ𝑒𝑛 𝐴−1 𝑒𝑥𝑖𝑠𝑡 𝑠𝑜, 𝐿. 𝐻. 𝑆 = 4𝐴 − 3𝐴
𝐴𝑋 = 𝐵 2 3 2 3
𝑝𝑟𝑒 𝑚𝑢𝑙𝑡𝑖𝑛𝑔 𝑏𝑦 𝐴−1 4[ ] − 3[ ]
1 5 1 5
𝐴 −1 (𝐴𝑋)
= 𝐴−1 𝐵 8 12 6 9
=[ ]−[ ]
(𝐴−1 𝐴)𝑋 = 𝐴−1 𝐵 4 20 3 15
8 − 6 12 − 9
𝐼2 𝑋 = 𝐴−1 𝐵 ∵ 𝐴−1 𝐴 = 𝐼2 =[ ]
4 − 3 20 − 15
𝑋 = 𝐴−1 𝐵 2 3
𝑥1 1 𝑎22 −𝑎12 𝑏1 =[ ]
Or [𝑥 ] = |𝐴| [−𝑎 1 5
2 21 𝑎11 ] [𝑏2 ] = 𝐴 = 𝑅. 𝐻. 𝑆 𝑝𝑟𝑜𝑣𝑒d
1 𝑎22 𝑏1 − 𝑎12 𝑏2 𝒊𝒊) 𝟑𝑩 − 𝟑𝑨 = 𝟑(𝑩 − 𝑨)
= [ ]
|𝐴| −𝑎21 𝑏1 + 𝑎11 𝑏2 Solution:
𝐿. 𝐻. 𝑆 𝟑𝑩 − 𝟑𝑨
𝑎22 𝑏1 − 𝑎12 𝑏2 𝟏 𝟕 𝟐 𝟑
= 3[ ]−𝟑[ ]
pk
|𝐴| 𝟔 𝟒 𝟏 𝟓
=[ ] 𝟑 𝟐𝟏 𝟔 𝟗
−𝑎21 𝑏1 + 𝑎11 𝑏2 =[ ]−[ ]
|𝐴| 𝟏𝟖 𝟏𝟐 𝟑 𝟏𝟓
𝑎22 𝑏1 −𝑎12 𝑏2
s. 𝟑−𝟔 𝟐𝟏 − 𝟗
𝑥1 = and =[ ]
|𝐴| 𝟏𝟖 − 𝟑 𝟏𝟐 − 𝟏𝟓
−𝑎21 𝑏1 +𝑎11 𝑏2 −𝟑 𝟏𝟐
𝑥2 = =[ ] → (𝟏)
te
|𝐴| 𝟏𝟓 𝟏𝟓
𝑡ℎ𝑢𝑠 𝑅. 𝐻. 𝑆
𝑏 𝑎12
| 1 | 𝟏 𝟕 𝟐 𝟑
no
𝑥1 =
𝑏2 𝑎22
and 𝟑(𝑩 − 𝑨) = 𝟑 [[ ]−[ ]]
|𝐴| 𝟔 𝟒 𝟏 𝟓
𝑎 𝑏1 𝟏−𝟐 𝟕−𝟑
| 11 | = 3 [[ ]]
2 𝑎12 𝑏2 𝟔−𝟏 𝟒−𝟓
𝑥 =
sy
|𝐴| −𝟏 𝟒
= 3[ ]
𝟓 −𝟏
−𝟑 𝟏𝟐
=[ ] → (𝟐)
ea
𝟏𝟓 𝟏𝟓
ℎ𝑒𝑛𝑐𝑒 𝐿. 𝐻. 𝑆 = 𝑅. 𝐻. 𝑆
Q2.
𝒊 𝟎
𝒊𝒇 𝑨 = [ ] 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝑨𝟒 = 𝑰𝟐
𝟏 −𝒊
Solution: 𝑵𝒐𝒕𝒆 𝒊 = √−𝟏 = 𝒊𝟐 = −𝟏
𝒊 𝟎 𝒊 𝟎
∵ 𝑨𝟐 = 𝑨 × 𝑨 = [ ]×[ ]
𝟏 −𝒊 𝟏 −𝒊
𝟐
𝑨𝟐 = [𝒊 + 𝟎 𝟎 − 𝟎𝟐 ]
𝒊−𝒊 𝟎+𝒊
−𝟏 𝟎
=[ ]
𝟎 −𝟏
−𝟏 𝟎 −𝟏 𝟎
𝑨𝟒 = 𝑨𝟐 × 𝑨𝟐 = [ ]×[ ]
𝟎 −𝟏 𝟎 −𝟏
𝟏+𝟎 𝟎+𝟎
𝑨𝟒 = [ ]
𝟎+𝟎 𝟎+𝟏
𝟏 𝟎
=[ ] = 𝑰𝟐 𝑯𝒆𝒏𝒄𝒆 𝑷𝒓𝒐𝒗𝒆𝒅
𝟎 𝟏
Q3.
𝒇𝒊𝒏𝒅 𝒙 𝒂𝒏𝒅 𝒚 𝒊𝒇
𝒙+𝟑 𝟏 𝟐 𝟏
[ ]=[ ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐
3|Page
, Class 11 Chapter 3
Solution: 4 2𝑥 𝑥 + 2𝑦 4 −2 3
[ ]=[ ]
𝒙+𝟑 𝟏 𝟐 𝟏 1 4+𝑦 1 1 6 1
[ ]=[ ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐 2𝑥 = −2 ⇒ 𝑥 = −1
𝒙+𝟑=𝟐 , 𝟑𝒚 − 𝟒 = 𝟐 4+𝑦 =6 ⇒𝑦 =2
𝒙 = 𝟐 − 𝟑 , 𝟑𝒚 = 𝟐 + 𝟒 𝑥 = −1 𝑎𝑛𝑑 𝑦 = 2
𝟔
𝒙 = −𝟏 , 𝒚 = 𝟑 = 𝟐 Q 6. 𝒊𝒇 𝑨 = [𝒂𝒊𝒋 ] 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕
𝟐×𝟐
So 𝒙 = −𝟏 𝒂𝒏𝒅 𝒚 = 𝟐 i) 𝝀(𝝁𝑨) = (𝝀𝝁)𝑨
ii) Solution :
𝒙+𝟑 𝟏 𝑎11 𝑎12 𝑎13
𝒚 𝟏
[ ]=[ ] 𝑳. 𝑯. 𝑺 = 𝝀(𝝁𝑨) 𝑨 = [𝑎21 𝑎22 𝑎23 ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐𝒙
Solution: 𝑎31 𝑎32 𝑎33
𝑎11 𝑎12 𝑎13
𝑥+3=𝑦 3𝑦 − 4 = 2𝑥
= 𝝀 ( 𝝁 [𝑎21 𝑎22 𝑎23 ] )
2𝑥 − 3𝑦 + 4 = 0 𝑎31 𝑎32 𝑎33
2𝑥 − 3(𝑥 + 3) + 4 = 0 𝜇𝑎11 𝜇𝑎12 𝜇𝑎13
2𝑥 − 3𝑥 − 9 + 4 = 0 = 𝝀 ( [𝜇𝑎21 𝜇𝑎22 𝜇𝑎23 ] )
−𝑥 − 5 = 0 𝜇𝑎31 𝜇𝑎32 𝜇𝑎33
−𝑥 = 5 𝜆𝜇𝑎11 𝜆𝜇𝑎12 𝜆𝜇𝑎13
𝑥 = −5 𝑝𝑢𝑡 𝑖𝑛(1) = [𝜆𝜇𝑎21 𝜆𝜇𝑎22 𝜆𝜇𝑎23 ]
−5 + 3 = 𝑦 𝜆𝜇𝑎31 𝜆𝜇𝑎32 𝜆𝜇𝑎33
𝑦 = −2
pk
So 𝑥 = −5 𝑎𝑛𝑑 𝑦 = −2 𝑎11 𝑎12 𝑎13
Q.4 = 𝜆𝜇 [𝑎21 𝑎22 𝑎23 ]
−𝟏 𝟐 𝟑 𝟎 𝟑 𝟐 s. 𝑎31 𝑎32 𝑎33
𝒊𝒇 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [ ] = (𝜆𝜇)𝐴 = 𝑅. 𝐻. 𝑆
𝟏 𝟎 𝟐 𝟏 −𝟏 𝟐
𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔 ℎ𝑒𝑛𝑐𝑒 𝑝𝑟𝑜𝑣𝑒𝑑.
te
(𝒊)𝟒𝑨 − 𝟑𝑩 (𝒊𝒊) 𝑨 + 𝟑(𝑩 − 𝑨) 𝒊𝒊) (𝝀 + 𝝁)𝑨 = 𝝀𝑨 + 𝝁𝑨
Solution: 𝐿. 𝐻. 𝑆
−𝟏 𝟐 𝟑 𝟎 𝟑 𝟐 𝑎11 𝑎12 𝑎13
no
𝟒𝑨 − 𝟑𝑩 = 𝟒 [ ]−𝟑[ ]
𝟏 𝟎 𝟐 𝟏 −𝟏 𝟐 = (𝜆 + 𝜇) [ 21 𝑎22 𝑎23 ]
𝑎
−4 8 12 0 9 6 𝑎31 𝑎32 𝑎33
=[ ]−[ ]
4 0 8 3 −3 6 (𝜆 + 𝜇)𝑎11 (𝜆 + 𝜇)𝑎12 (𝜆 + 𝜇)𝑎13
−4 − 0 8 − 9 12 − 6
sy
=[ ] = [(𝜆 + 𝜇)𝑎21 (𝜆 + 𝜇)𝑎22 (𝜆 + 𝜇)𝑎23 ]
4−3 0+3 8−6
−4 −1 6 (𝜆 + 𝜇)𝑎31 (𝜆 + 𝜇)𝑎32 (𝜆 + 𝜇)𝑎33
4𝐴 − 3𝐵 = [ ] 𝜆𝑎11 + 𝜇𝑎11 𝜆𝑎12 + 𝜇𝑎12 𝜆𝑎13 + 𝜇𝑎13
1 3 2
ea
ii) = [𝜆𝑎21 + 𝜇𝑎21 𝜆𝑎22 + 𝜇𝑎22 𝜆𝑎23 + 𝜇𝑎23 ]
𝐴 + 3(𝐵 − 𝐴) 𝜆𝑎31 + 𝜇𝑎31 𝜆𝑎32 + 𝜇𝑎32 𝜆𝑎33 + 𝜇𝑎33
−1 2 3 0 3 2 −1 2 3 𝜆𝑎11 𝜆𝑎12 𝜆𝑎13 𝜇𝑎11 𝜇𝑎12 𝜇𝑎13
=[ ] + 3 ([ ]−[ ])
1 0 2 1 −1 2 1 0 2 = [𝜆𝑎21 𝜆𝑎22 𝜆𝑎23 ] + [ 21 𝜇𝑎22 𝜇𝑎23 ]
𝜇𝑎
−1 2 3 0+1 3−2 2−3 𝜆𝑎31 𝜆𝑎32 𝜆𝑎33 𝜇𝑎31 𝜇𝑎32 𝜇𝑎33
=[ ] + 3 ([ ])
1 0 2 1 − 1 −1 − 0 2 − 2 𝑎11 𝑎12 𝑎13 𝑎11 𝑎12 𝑎13
−1 2 3 1 1 −1 𝑎 𝑎 𝑎
=[ ] + 3[ ] = 𝜆 [ 21 22 23 ] + 𝝁 [ 21 𝑎22 𝑎23 ]
𝑎
1 0 2 0 −1 0 𝑎31 𝑎32 𝑎33 𝑎31 𝑎32 𝑎33
−1 2 3 3 3 −3
=[ ]+[ ] = 𝜆𝐴 + 𝜇𝐴 = 𝑅. 𝐻. 𝑆 ℎ𝑒𝑛𝑐𝑒 𝑝𝑟𝑜𝑣𝑒𝑑
1 0 2 0 −3 0
−1 + 3 2 + 3 3 − 3 (𝒊𝒊𝒊)𝝀𝑨 − 𝑨 = (𝝀 − 𝟏)𝑨
=[ ] 𝑎11 𝑎12 𝑎13 𝑎11 𝑎12 𝑎13
1+0 0−3 2−0
=[
2 5 0
] 𝝀𝑨 − 𝑨 = 𝝀 [𝑎21 𝑎22 𝑎23 ] − [𝑎21 𝑎22 𝑎23 ]
1 −3 2 𝑎31 𝑎32 𝑎33 𝑎31 𝑎32 𝑎33
𝜆𝑎11 𝜆𝑎12 𝜆𝑎13 𝑎 11 𝑎12 𝑎13
Q5. 𝑭𝒊𝒏𝒅 𝒙 𝒂𝒏𝒅 𝒚 𝒊𝒇 = [𝜆𝑎21 𝜆𝑎22 𝜆𝑎23 ] − [𝑎21 𝑎22 𝑎23 ]
𝟐 𝟎 𝒙 𝟏 𝒙 𝒚 𝟒 −𝟐 𝟑 𝜆𝑎31 𝜆𝑎32 𝜆𝑎33 𝑎31 𝑎32 𝑎33
[ ]+𝟐[ ]=[ ]
𝟏 𝒚 𝟑 𝟎 𝟐 −𝟏 𝟏 𝟔 𝟏 𝜆𝑎11 − 𝑎11 𝜆𝑎12 − 𝑎12 𝜆𝑎13− 𝑎13
Solution: = [𝜆𝑎21 − 𝑎21 𝜆𝑎22 − 𝑎22 𝜆𝑎23 − 𝑎23 ]
2 0 𝑥 2 2𝑥 2𝑦 4 −2 3 𝜆𝑎31 − 𝑎31 𝜆𝑎32 − 𝑎32 𝜆𝑎33 − 𝑎33
[ ]+[ ]=[ ]
1 𝑦 3 0 4 −2 1 6 1
2 + 2 0 + 2𝑥 𝑥 + 2𝑦 4 −2 3
[ ]=[ ]
1+0 𝑦+4 3−2 1 6 1
4|Page
3.1 Introduction *row matrix and column matrix are also called row
Matrix: vector and column.
A rectangular array of numbers enclosed by a pair Principal Diagonal:
of bracket is called a matrix. The diagonal from upper left corner to the lower
2 −1 3 right corner of a square matrix is called principal or
𝑒. 𝑔; [ ]
−5 4 7 main diagonal or leading diagonal.
Note: e.g.;
i) Matrices are denoted by capital letters 𝑎11 𝑎12 𝑎13
such as A,B,C…,X,Y,Z
[ 21 𝑎22 𝑎23 ]
𝑎
ii) The elements or entries of a matrix as 𝑎31 𝑎32 𝑎33
denoted by small letters such as
𝑎, 𝑏, 𝑐, … , 𝑥, 𝑦, 𝑧. the entries 𝑎11 , 𝑎12 , 𝑎13 form the principal diagonal
iii) The horizontal lines of elements are Secondary diagonal:
called rows of a matrix. The diagonal from lower left corner to the upper
iv) The vertical lines of elements are called right corner of a square matrix is called secondary
columns of a matrix. diagonal or leading diagonal.
Order of a matrix. e.g.;
The number of rows and columns of a matrix is 𝑎11 𝑎12 𝑎13
called order of a matrix. [𝑎21 𝑎22 𝑎23 ]
𝑎31 𝑎32 𝑎33
pk
i.e.; if a matrix has m rows and n column then its
order is 𝑚 × 𝑛 (read as m-by-n). The entries 𝑎13 , 𝑎22 , 𝑎𝑛𝑑 𝑎31 𝑓𝑜𝑟𝑚 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦
2 −1 3
𝐴=[ ] 𝑖𝑡𝑠 𝑜𝑟𝑑𝑒𝑟 𝑖𝑠 2 × 3 Diagonal.
1 4 7 s.
𝐵 = [1 4 6] 𝑖𝑡𝑠 𝑜𝑟𝑑𝑒𝑟 𝑖𝑠 1 × 3 Diagonal Matrix:
*the matrix A is called real if all of its elements are A square matrix in which all elements except the
te
real. diagonal are zero is called a diagonal matrix.
Types of Matrices 3 0 0
1 0
e.g.; [ ],[0 3 0]
no
Row Matrix: 0 2
A matrix which has only one row i.e; a matrix of 0 0 3
Identity Matrix:
order 1 × 𝑛 is called row matrix. A diagonal matrix whose all elements of the main
e.g.;[𝑎11 𝑎12 𝑎13 ] ,
sy
diagonal are 1 is called identity matrix denoted by
[1 2 3 ] etc. 𝐼𝑛
Column Matrix: 1 0 0
A matrix which has only one column. i.e.; a matrix 1 0
e.g.; [ ] ,[0 1 0]
ea
of order 𝑚 × 1 is called column matrix. e.g.; 0 1 2×2
0 0 1 3×3
𝑎11 1
[𝑎12 ] , [2] 𝑒𝑡𝑐. Scalar Matrix:
𝑎13 3 A diagonal matrix whose all elements of the main
Rectangular Matrix.
diagonal are same is called scalar matrix.
A matrix whose number of rows and columns are 𝑘 0 0
not equal is called rectangular matrix.eg.; 2 0
e.g.; [ ] ,[0 𝑘 0]
2 −1 3 𝑎11 𝑎12 𝑎13 0 2 2×2
[ ] , [𝑎 ] 0 0 𝑘 3×3
1 4 7 @1 𝑎22 𝑎23
Square Matrix: Needs to remember
A matrix whose number of rows and columns are Square Matrix: A matrix having m rows and n
equal is called square matrix. e.g.; columns with m=n is called square matrix.
1 2 𝑎11 𝑎12 Rectangular Matrix: A matrix having m rows and n
[ ] 𝑎𝑛𝑑 [𝑎 ] columns with 𝑚 ≠ 𝑛 is called rectangular matrix.
3 4 21 𝑎22
Null Matrix: Diagonal Matrix: let 𝐴 = [𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥
A matrix whose all elements are zero is called null of order n, if 𝑎𝑖𝑗 = 0∀ 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 𝑎𝑖𝑗 ≠
matrix. 0 𝑓𝑜𝑟
0 0 0 0 0 𝑖 = 𝑗 𝑠𝑜𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙
e.g.; [ ], [ ]
0 0 0 0 0 𝑜𝑓 𝐴 𝑚𝑎𝑦 𝑏𝑒 𝑧𝑒𝑟𝑜 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑎𝑙𝑙, 𝑡ℎ𝑒𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 𝑖𝑠
𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑚𝑎𝑡𝑟𝑖𝑥.
1|Page
,Class 11 Chapter 3
Identity Matrix: let 𝐴 = If the product AB is defined, then the order
[𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 of the product can be illustrated as given
𝑜𝑟𝑑𝑒𝑟 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑖𝑗 = 1 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 = 𝑗 below.
Then A is called unit matrix or identity matrix Order of A 𝑚×𝑛
denoted by 𝐼𝑛 . 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝐵 𝑛×𝑝
Null Matrix: A matrix of order 𝑚 × 𝑛 with all Order of AB 𝑚×𝑛
elements zero is called null matrix. NOTE:
Scalar Matrix: let 𝐴 = Powers of square matrices are defined as 𝐴2 = 𝐴 ×
[𝑎𝑖𝑗 ]𝑏𝑒 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝐴
Order n. if 𝑎𝑖𝑗 = 0 ∀ 𝑖 ≠ 𝑗 𝑎𝑛𝑑 𝑎𝑖𝑗 = 𝑘(𝑠𝑜𝑚𝑒 𝑛𝑜𝑛 𝐴3 = 𝐴 × 𝐴 × 𝐴 … … 𝑡𝑜 𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑠.
Determinant of 𝟐 × 𝟐 Matrix:
𝑧𝑒𝑟𝑜 𝑠𝑐𝑎𝑙𝑎𝑟) ∀ 𝑖 = 𝑗 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑠𝑜𝑚𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑖𝑠
𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑐𝑎𝑙𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥. The determinant of a matrix is denoted by enclosing
Equal matrices: its square array between vertical bars instead of
Two matrices of the same order are said to be equal brackets.
𝑎 𝑐 𝑎 𝑐
if corresponding entries are equal. e.g; 𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
𝑏 𝑑 𝑏 𝑑
Addition of Matrices:
Two matrices can be added if both have same |𝐴| = 𝑎𝑑 − 𝑏𝑐
order. 𝑓𝑜𝑟 𝑒𝑥𝑎𝑚𝑝𝑙
*addition is done by adding corresponding entries 2 −1 2 −1
𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
pk
of the matrices. 4 3 4 3
*Matrices of different orders cannot be added. |𝐴| = (2)(3) − (4)(−1) = 6 + 4 = 10
Transpose of matrix: Hence the determinant of a matrix is the difference
s.
Transpose of a matrix is denoted by 𝐴𝑡 can be of the product of the entries in the two diagonals.
obtained by interchanging rows into column or Singular Matrix:
A square matrix A is said to be singular if |𝐴| = 0
te
column into rows.
Scalar multiplication: Example:
8 4 8 4
𝑙𝑒𝑡 𝐴 = [𝑎𝑖𝑗 ]𝑏𝑒 𝑎𝑛 𝑚 × 𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
no
2 1 2 1
𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑛𝑑 𝑘 𝑏𝑒 𝑎 𝑠𝑐𝑙𝑎𝑟 |𝐴| = (8)(1) − (4)(2) = 8 − 8 = 0
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑘 𝑘 𝑎𝑛𝑑 𝐴 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 Non-Singular Matrix:
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓𝐴 𝑏𝑦 𝑘. A square matrix A is said to be Non- singular if |𝐴| ≠
sy
𝑖. 𝑒 𝑘𝐴 = [𝑘𝑎𝑖𝑗 ] 0
𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝑘𝐴 𝑖𝑠 𝑚 × 𝑛 Example:
*𝑖𝑓 𝑛 𝑖𝑠 𝑎 + 𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 , 1 2 1 2
ea
𝑖𝑓 𝐴 = [ ] 𝑡ℎ𝑒𝑛 |𝐴| = | |
𝑡ℎ𝑒𝑛 𝐴 + 𝐴 + ⋯ , 𝑡𝑜 𝑛𝑡𝑒𝑟𝑚𝑠 = 𝑛𝐴 4 9 4 9
Subtraction of matrices: |𝐴| = (1)(9) − (2)(4) = 9 − 8
𝑖𝑓 𝐴 = [𝑎𝑖𝑗 ]𝑎𝑛𝑑 𝐵 = [𝑏𝑖𝑗 ] 𝑎𝑟𝑒 𝑚𝑎𝑡𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 = 1 ≠ 0 𝑠𝑜 𝐴 𝑖𝑠 𝑛𝑜𝑛 − 𝑠𝑖𝑛𝑔𝑢𝑙𝑎𝑟
Ad joint of 𝟐 × 𝟐 𝑴𝒂𝒕𝒓𝒊𝒙:
𝑚 × 𝑛, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑑𝑒𝑓𝑖𝑛𝑒 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐵 𝑓𝑟𝑜𝑚 𝐴
𝑎 𝑏
As: 𝑡ℎ𝑒 𝑎𝑑𝑗𝑜𝑖𝑛𝑡 𝑜𝑓 𝑎 𝑚𝑎𝑡𝑟𝑖𝑥 𝐴 = [ ] 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑
𝑐 𝑑
𝐴 − 𝐵 = 𝐴 + (−𝐵) = [𝑎𝑖𝑗 ] + [−𝑏𝑖𝑗 ] 𝑑 −𝑏
𝑏𝑦 𝑎𝑑𝑗𝐴 𝑎𝑛𝑑 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠 𝑎𝑑𝑗𝐴 = [ ]
𝐴 − 𝐵 = [𝑎𝑖𝑗 − 𝑏𝑖𝑗 ] −𝑐 𝑎
𝑖 = 1,2,3, . . , 𝑚, 𝑗 = 1,2,3, … , 𝑛 Inverse of 𝟐 × 𝟐 𝑴𝒂𝒕𝒓𝒊𝒙
𝐴 Let A be a non-singular square matrix of order 2. If
− 𝑏 𝑖𝑓 𝑓𝑜𝑟𝑚𝑒𝑑 𝑏𝑦 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑒𝑎𝑐ℎ 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓 𝐵 there exist of matrix B such that
𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑡𝑟𝑦 𝑜𝑓 𝐴. 1 0
𝐴𝐵 = 𝐵𝐴 = 𝐼2 𝑤ℎ𝑒𝑟𝑒 𝐼2 = [ ] 𝑡ℎ𝑒𝑛 𝐵 𝑖𝑠
Multiplication of two Matrix: 0 1
𝑐𝑎𝑙𝑙𝑒𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝐴 𝑎𝑛𝑑 𝑖𝑠 𝑢𝑠𝑢𝑎𝑙𝑙𝑦
Two matrix A and B are said to be conformable for
𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴−1
product AB if the number of columns of A is equal i.e.; 𝐵 = 𝐴−1 𝑡ℎ𝑢𝑠
to the number of rows of B. 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼2
*Matrix multiplication is not commutative 𝑖. 𝑒;. Solution Of Simultaneous Linear Equations By
𝐴𝐵 ≠ 𝐵𝐴 Using Matrices:
𝑙𝑒𝑡 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑒𝑞𝑠. 𝑏𝑒
𝑎11 𝑥1 + 𝑎12 𝑥2 = 𝑏1
2|Page
,Class 11 Chapter 3
𝑎21 𝑥1 + 𝑎22 𝑥2 = 𝑏2 Exercise 3.1
𝑤ℎ𝑒𝑟𝑒 𝑎11 , 𝑎12 , 𝑎21 , 𝑎22 , 𝑏1 , 𝑏2 ∈ 𝑅 𝟐 𝟑 𝟏 𝟕
Given system in matrix form Q1. If 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [ ]
𝟏 𝟓 𝟔 𝟒
𝑎11 𝑎12 𝑥1 𝑏 𝒕𝒉𝒆𝒏 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝒊) 𝟒𝑨 − 𝟑𝑨 = 𝑨
[𝑎 𝑎 ] [𝑥 ] = [ 1 ]
21 22 2 𝑏2 𝒊𝒊) 𝟑𝑩 − 𝟑𝑨 = 𝟑(𝑩 − 𝑨)
𝑎11 𝑎12 𝑥1 𝑏
𝑙𝑒𝑡 𝐴 = [𝑎 ] , 𝑋 = [𝑥 ] , 𝐵 = [ 1 ] Solution:
21 𝑎22 2 𝑏2 4𝐴 − 3𝐴 = 𝐴
𝑖𝑓|𝐴| ≠ 0 𝑡ℎ𝑒𝑛 𝐴−1 𝑒𝑥𝑖𝑠𝑡 𝑠𝑜, 𝐿. 𝐻. 𝑆 = 4𝐴 − 3𝐴
𝐴𝑋 = 𝐵 2 3 2 3
𝑝𝑟𝑒 𝑚𝑢𝑙𝑡𝑖𝑛𝑔 𝑏𝑦 𝐴−1 4[ ] − 3[ ]
1 5 1 5
𝐴 −1 (𝐴𝑋)
= 𝐴−1 𝐵 8 12 6 9
=[ ]−[ ]
(𝐴−1 𝐴)𝑋 = 𝐴−1 𝐵 4 20 3 15
8 − 6 12 − 9
𝐼2 𝑋 = 𝐴−1 𝐵 ∵ 𝐴−1 𝐴 = 𝐼2 =[ ]
4 − 3 20 − 15
𝑋 = 𝐴−1 𝐵 2 3
𝑥1 1 𝑎22 −𝑎12 𝑏1 =[ ]
Or [𝑥 ] = |𝐴| [−𝑎 1 5
2 21 𝑎11 ] [𝑏2 ] = 𝐴 = 𝑅. 𝐻. 𝑆 𝑝𝑟𝑜𝑣𝑒d
1 𝑎22 𝑏1 − 𝑎12 𝑏2 𝒊𝒊) 𝟑𝑩 − 𝟑𝑨 = 𝟑(𝑩 − 𝑨)
= [ ]
|𝐴| −𝑎21 𝑏1 + 𝑎11 𝑏2 Solution:
𝐿. 𝐻. 𝑆 𝟑𝑩 − 𝟑𝑨
𝑎22 𝑏1 − 𝑎12 𝑏2 𝟏 𝟕 𝟐 𝟑
= 3[ ]−𝟑[ ]
pk
|𝐴| 𝟔 𝟒 𝟏 𝟓
=[ ] 𝟑 𝟐𝟏 𝟔 𝟗
−𝑎21 𝑏1 + 𝑎11 𝑏2 =[ ]−[ ]
|𝐴| 𝟏𝟖 𝟏𝟐 𝟑 𝟏𝟓
𝑎22 𝑏1 −𝑎12 𝑏2
s. 𝟑−𝟔 𝟐𝟏 − 𝟗
𝑥1 = and =[ ]
|𝐴| 𝟏𝟖 − 𝟑 𝟏𝟐 − 𝟏𝟓
−𝑎21 𝑏1 +𝑎11 𝑏2 −𝟑 𝟏𝟐
𝑥2 = =[ ] → (𝟏)
te
|𝐴| 𝟏𝟓 𝟏𝟓
𝑡ℎ𝑢𝑠 𝑅. 𝐻. 𝑆
𝑏 𝑎12
| 1 | 𝟏 𝟕 𝟐 𝟑
no
𝑥1 =
𝑏2 𝑎22
and 𝟑(𝑩 − 𝑨) = 𝟑 [[ ]−[ ]]
|𝐴| 𝟔 𝟒 𝟏 𝟓
𝑎 𝑏1 𝟏−𝟐 𝟕−𝟑
| 11 | = 3 [[ ]]
2 𝑎12 𝑏2 𝟔−𝟏 𝟒−𝟓
𝑥 =
sy
|𝐴| −𝟏 𝟒
= 3[ ]
𝟓 −𝟏
−𝟑 𝟏𝟐
=[ ] → (𝟐)
ea
𝟏𝟓 𝟏𝟓
ℎ𝑒𝑛𝑐𝑒 𝐿. 𝐻. 𝑆 = 𝑅. 𝐻. 𝑆
Q2.
𝒊 𝟎
𝒊𝒇 𝑨 = [ ] 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝑨𝟒 = 𝑰𝟐
𝟏 −𝒊
Solution: 𝑵𝒐𝒕𝒆 𝒊 = √−𝟏 = 𝒊𝟐 = −𝟏
𝒊 𝟎 𝒊 𝟎
∵ 𝑨𝟐 = 𝑨 × 𝑨 = [ ]×[ ]
𝟏 −𝒊 𝟏 −𝒊
𝟐
𝑨𝟐 = [𝒊 + 𝟎 𝟎 − 𝟎𝟐 ]
𝒊−𝒊 𝟎+𝒊
−𝟏 𝟎
=[ ]
𝟎 −𝟏
−𝟏 𝟎 −𝟏 𝟎
𝑨𝟒 = 𝑨𝟐 × 𝑨𝟐 = [ ]×[ ]
𝟎 −𝟏 𝟎 −𝟏
𝟏+𝟎 𝟎+𝟎
𝑨𝟒 = [ ]
𝟎+𝟎 𝟎+𝟏
𝟏 𝟎
=[ ] = 𝑰𝟐 𝑯𝒆𝒏𝒄𝒆 𝑷𝒓𝒐𝒗𝒆𝒅
𝟎 𝟏
Q3.
𝒇𝒊𝒏𝒅 𝒙 𝒂𝒏𝒅 𝒚 𝒊𝒇
𝒙+𝟑 𝟏 𝟐 𝟏
[ ]=[ ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐
3|Page
, Class 11 Chapter 3
Solution: 4 2𝑥 𝑥 + 2𝑦 4 −2 3
[ ]=[ ]
𝒙+𝟑 𝟏 𝟐 𝟏 1 4+𝑦 1 1 6 1
[ ]=[ ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐 2𝑥 = −2 ⇒ 𝑥 = −1
𝒙+𝟑=𝟐 , 𝟑𝒚 − 𝟒 = 𝟐 4+𝑦 =6 ⇒𝑦 =2
𝒙 = 𝟐 − 𝟑 , 𝟑𝒚 = 𝟐 + 𝟒 𝑥 = −1 𝑎𝑛𝑑 𝑦 = 2
𝟔
𝒙 = −𝟏 , 𝒚 = 𝟑 = 𝟐 Q 6. 𝒊𝒇 𝑨 = [𝒂𝒊𝒋 ] 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕
𝟐×𝟐
So 𝒙 = −𝟏 𝒂𝒏𝒅 𝒚 = 𝟐 i) 𝝀(𝝁𝑨) = (𝝀𝝁)𝑨
ii) Solution :
𝒙+𝟑 𝟏 𝑎11 𝑎12 𝑎13
𝒚 𝟏
[ ]=[ ] 𝑳. 𝑯. 𝑺 = 𝝀(𝝁𝑨) 𝑨 = [𝑎21 𝑎22 𝑎23 ]
−𝟑 𝟑𝒚 − 𝟒 −𝟑 𝟐𝒙
Solution: 𝑎31 𝑎32 𝑎33
𝑎11 𝑎12 𝑎13
𝑥+3=𝑦 3𝑦 − 4 = 2𝑥
= 𝝀 ( 𝝁 [𝑎21 𝑎22 𝑎23 ] )
2𝑥 − 3𝑦 + 4 = 0 𝑎31 𝑎32 𝑎33
2𝑥 − 3(𝑥 + 3) + 4 = 0 𝜇𝑎11 𝜇𝑎12 𝜇𝑎13
2𝑥 − 3𝑥 − 9 + 4 = 0 = 𝝀 ( [𝜇𝑎21 𝜇𝑎22 𝜇𝑎23 ] )
−𝑥 − 5 = 0 𝜇𝑎31 𝜇𝑎32 𝜇𝑎33
−𝑥 = 5 𝜆𝜇𝑎11 𝜆𝜇𝑎12 𝜆𝜇𝑎13
𝑥 = −5 𝑝𝑢𝑡 𝑖𝑛(1) = [𝜆𝜇𝑎21 𝜆𝜇𝑎22 𝜆𝜇𝑎23 ]
−5 + 3 = 𝑦 𝜆𝜇𝑎31 𝜆𝜇𝑎32 𝜆𝜇𝑎33
𝑦 = −2
pk
So 𝑥 = −5 𝑎𝑛𝑑 𝑦 = −2 𝑎11 𝑎12 𝑎13
Q.4 = 𝜆𝜇 [𝑎21 𝑎22 𝑎23 ]
−𝟏 𝟐 𝟑 𝟎 𝟑 𝟐 s. 𝑎31 𝑎32 𝑎33
𝒊𝒇 𝑨 = [ ] 𝒂𝒏𝒅 𝑩 = [ ] = (𝜆𝜇)𝐴 = 𝑅. 𝐻. 𝑆
𝟏 𝟎 𝟐 𝟏 −𝟏 𝟐
𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒇𝒐𝒍𝒍𝒐𝒘𝒊𝒏𝒈 𝒎𝒂𝒕𝒓𝒊𝒄𝒆𝒔 ℎ𝑒𝑛𝑐𝑒 𝑝𝑟𝑜𝑣𝑒𝑑.
te
(𝒊)𝟒𝑨 − 𝟑𝑩 (𝒊𝒊) 𝑨 + 𝟑(𝑩 − 𝑨) 𝒊𝒊) (𝝀 + 𝝁)𝑨 = 𝝀𝑨 + 𝝁𝑨
Solution: 𝐿. 𝐻. 𝑆
−𝟏 𝟐 𝟑 𝟎 𝟑 𝟐 𝑎11 𝑎12 𝑎13
no
𝟒𝑨 − 𝟑𝑩 = 𝟒 [ ]−𝟑[ ]
𝟏 𝟎 𝟐 𝟏 −𝟏 𝟐 = (𝜆 + 𝜇) [ 21 𝑎22 𝑎23 ]
𝑎
−4 8 12 0 9 6 𝑎31 𝑎32 𝑎33
=[ ]−[ ]
4 0 8 3 −3 6 (𝜆 + 𝜇)𝑎11 (𝜆 + 𝜇)𝑎12 (𝜆 + 𝜇)𝑎13
−4 − 0 8 − 9 12 − 6
sy
=[ ] = [(𝜆 + 𝜇)𝑎21 (𝜆 + 𝜇)𝑎22 (𝜆 + 𝜇)𝑎23 ]
4−3 0+3 8−6
−4 −1 6 (𝜆 + 𝜇)𝑎31 (𝜆 + 𝜇)𝑎32 (𝜆 + 𝜇)𝑎33
4𝐴 − 3𝐵 = [ ] 𝜆𝑎11 + 𝜇𝑎11 𝜆𝑎12 + 𝜇𝑎12 𝜆𝑎13 + 𝜇𝑎13
1 3 2
ea
ii) = [𝜆𝑎21 + 𝜇𝑎21 𝜆𝑎22 + 𝜇𝑎22 𝜆𝑎23 + 𝜇𝑎23 ]
𝐴 + 3(𝐵 − 𝐴) 𝜆𝑎31 + 𝜇𝑎31 𝜆𝑎32 + 𝜇𝑎32 𝜆𝑎33 + 𝜇𝑎33
−1 2 3 0 3 2 −1 2 3 𝜆𝑎11 𝜆𝑎12 𝜆𝑎13 𝜇𝑎11 𝜇𝑎12 𝜇𝑎13
=[ ] + 3 ([ ]−[ ])
1 0 2 1 −1 2 1 0 2 = [𝜆𝑎21 𝜆𝑎22 𝜆𝑎23 ] + [ 21 𝜇𝑎22 𝜇𝑎23 ]
𝜇𝑎
−1 2 3 0+1 3−2 2−3 𝜆𝑎31 𝜆𝑎32 𝜆𝑎33 𝜇𝑎31 𝜇𝑎32 𝜇𝑎33
=[ ] + 3 ([ ])
1 0 2 1 − 1 −1 − 0 2 − 2 𝑎11 𝑎12 𝑎13 𝑎11 𝑎12 𝑎13
−1 2 3 1 1 −1 𝑎 𝑎 𝑎
=[ ] + 3[ ] = 𝜆 [ 21 22 23 ] + 𝝁 [ 21 𝑎22 𝑎23 ]
𝑎
1 0 2 0 −1 0 𝑎31 𝑎32 𝑎33 𝑎31 𝑎32 𝑎33
−1 2 3 3 3 −3
=[ ]+[ ] = 𝜆𝐴 + 𝜇𝐴 = 𝑅. 𝐻. 𝑆 ℎ𝑒𝑛𝑐𝑒 𝑝𝑟𝑜𝑣𝑒𝑑
1 0 2 0 −3 0
−1 + 3 2 + 3 3 − 3 (𝒊𝒊𝒊)𝝀𝑨 − 𝑨 = (𝝀 − 𝟏)𝑨
=[ ] 𝑎11 𝑎12 𝑎13 𝑎11 𝑎12 𝑎13
1+0 0−3 2−0
=[
2 5 0
] 𝝀𝑨 − 𝑨 = 𝝀 [𝑎21 𝑎22 𝑎23 ] − [𝑎21 𝑎22 𝑎23 ]
1 −3 2 𝑎31 𝑎32 𝑎33 𝑎31 𝑎32 𝑎33
𝜆𝑎11 𝜆𝑎12 𝜆𝑎13 𝑎 11 𝑎12 𝑎13
Q5. 𝑭𝒊𝒏𝒅 𝒙 𝒂𝒏𝒅 𝒚 𝒊𝒇 = [𝜆𝑎21 𝜆𝑎22 𝜆𝑎23 ] − [𝑎21 𝑎22 𝑎23 ]
𝟐 𝟎 𝒙 𝟏 𝒙 𝒚 𝟒 −𝟐 𝟑 𝜆𝑎31 𝜆𝑎32 𝜆𝑎33 𝑎31 𝑎32 𝑎33
[ ]+𝟐[ ]=[ ]
𝟏 𝒚 𝟑 𝟎 𝟐 −𝟏 𝟏 𝟔 𝟏 𝜆𝑎11 − 𝑎11 𝜆𝑎12 − 𝑎12 𝜆𝑎13− 𝑎13
Solution: = [𝜆𝑎21 − 𝑎21 𝜆𝑎22 − 𝑎22 𝜆𝑎23 − 𝑎23 ]
2 0 𝑥 2 2𝑥 2𝑦 4 −2 3 𝜆𝑎31 − 𝑎31 𝜆𝑎32 − 𝑎32 𝜆𝑎33 − 𝑎33
[ ]+[ ]=[ ]
1 𝑦 3 0 4 −2 1 6 1
2 + 2 0 + 2𝑥 𝑥 + 2𝑦 4 −2 3
[ ]=[ ]
1+0 𝑦+4 3−2 1 6 1
4|Page
