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Mathematics Notes for Class 9thKPK All Boards
Mathematics
Chapter # 1
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Chapter # 1
Matrices
Exercise# 1.1 Examples
2 5
Matrix D=
1 3
A matrix is a rectangular array
In this example
(arrangements) of real numbers enclosed in
2, 5,1,3 all are the elements of a matrix D.
square brackets. Each number in a matrix
2, 5 and 1, 3 are the rows of a matrix D.
is called an element or entry of the matrix.
2, 1 and 5, 3 are the columns of matrix D.
Matrices are mostly denoted by capital
As No. of Rows= 2
letters.
And No. of Columns= 2
Examples
So order is 2−𝑏𝑦 − 2 (OR) 2 × 2
1 0 0
2 3 Equal Matrix
A= , C= 0 2 0
When two matrices of the same order and
0 5 0 0 1 the corresponding elements are same.
Exercise # 1.1
Rows and Columns of a Matrix Page # 6
The rows of a matrix run horizontally, Q1: Which of the following are square and
and the columns of a matrix run vertically. which are rectangular matrices?
Order or Dimension of a Matrix 2 3
(i) A=
The number of rows and columns that 0 5
a matrix has is called order of a matrix. As No. of Rows = No. of Columns
Order of a matrix is represented by: So it is Square matrix.
Order of matrix = 𝑚 × 𝑛
OR 6 3 −1
B=
Order of matrix = m-by-n (ii)
1 5 2
Here “m” represents number of Rows As No. of Rows ≠ No. of Columns
And “n” represents number of columns So it is Rectangular matrix.
Note
Order of a matrix is also called
1 0 0
dimension or size of a matrix.
(iii) C = 0 2 0
Examples 0 0 1
2 5 As No. of Rows = No. of Columns
D= So it is Square matrix.
1 3
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(iv) D = −5 (v) E = 3 2
As No. of Rows = No. of Columns As No. of Rows= 1
So it is Square matrix. And No. of Columns= 2
So order is 1−𝑏𝑦 − 2 (OR) 1× 2
(v) E = −3 4
1 2 3
As No. of Rows ≠ No. of Columns
So it is Rectangular matrix.
(vi) D = 6 5 9
0 0 0
(vi) −1 As No. of Rows= 3
F =
7 And No. of Columns= 3
As No. of Rows ≠ No. of Columns So order is 3−𝑏𝑦 − 3 (OR) 3× 3
So it is Rectangular matrix.
3 2 −4
Q2: List the order of the following matrices. −2 5 0
If = , give the following
1 2 −1 Q3: A
2 1 5
(i) A=
3 4 2
−3 4 6
As No. of Rows= 2
elements.
And No. of Columns= 3
Solution
So order is 2 − 𝑏𝑦 − 3 (OR) 2 × 3
3 2 −4
(ii) B = −4 −2 5 0
A=
As No. of Rows= 1 2 1 5
And No. of Columns= 1
−3 4 6
So order is 1−𝑏𝑦 − 1 (OR) 1× 1
a11 a12 a13
2 3 −1 a a22 a23
(iii) C= As
21
a31 a32
1 2 5 a33
As No. of Rows= 2 a41 a42 a43
And No. of Columns= 3
So order is 2 − 𝑏𝑦 − 3 (OR) 2 × 3 Answers:
(i) 𝑎12 = 2
(ii) 𝑎23 = 0
2 1
(iv) (iii) 𝑎32 = 1
F = 3 2 (iv) 𝑎43 = 6
4 −1 (v) 𝑎13 = −4
As No. of Rows= 3 (vi) 𝑎43 = 6
And No. of Columns= 2
So order is 3−𝑏𝑦 − 2(OR) 3× 2
, 3
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Q4: Which of the following matrices are Q6: If
equal? x + 3 z + 4 2 y − 7 0 6 3 y − 2
2 5 2 5 −6 a − 1
0 = −6 −3 2c + 2
A= , B= ,
1 3 4 3 b − 3 −21 0 2b + 4 −21 0
1 + 1 3 + 2 2 4 + 1 find the values of a, b, c, 𝒙, y and 𝒛.
C= , D=
2 + 1 3
Solution:
4 1
As
Solution: x + 3 z + 4 2 y − 7 0 6 3 y − 2
−6 a − 1
0 = −6 −3 2c + 2
2 5 2 5
As C = and D = b − 3 −21 0 2b + 4 −21 0
4 3 1 3
So A and D are equal i.e. 𝐴 = 𝐷 Now compare the corresponding elements
And B and C are equal i.e. 𝐵 = 𝐶 𝑥+3 =0
𝑥 = −3
Now
2 −3 v −3 𝑧+4=6
Q5: Let A = and B = , for 𝑧 = 6−4
u 0 5 w
𝑧=2
what values of 𝒖, 𝒗, 𝒂𝒏𝒅 𝒘 are when A
Now
and B equal.
2𝑦 − 7 = 3𝑦 − 2
Solution
−7 + 2 = 3𝑦 − 2𝑦
2 −3 v −3
A= , B= −5 = 𝑦
u 0 5 w 𝑦 = −5
As A and B are equal. So Now
2 −3 v −3 𝑎 − 1 = −3
u 0 = 5 w 𝑎 = −3 + 1
𝑎 = −2
Now compare the corresponding Now
elements 0 = 2𝑐 + 2
2=𝑣 0 − 2 = 2𝑐
Or 𝒗 = 𝟐 −2 = 2𝑐 Answers:
𝒖=𝟓 −2 𝑎 = −2
=𝑐
0=𝑤 2
𝑏 = −7
−1 = 𝑐
Or 𝒘 = 𝟎 𝑐 = −1
𝑐 = −1
Now 𝑥 = −3
𝑏 − 3 = 2𝑏 + 4 𝑦 = −5
−3 − 4 = 2𝑏 − 𝑏
𝑧=2
−7 = 𝑏
𝑏 = −7
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