Parents can use OP Malhotra Class 10 Solutions Chapter 8 Matrices Exercise 8(a) to provide additional support to their children.
S Chand Class 10 ICSE Maths Solutions Chapter 8 Matrices Exercise 8(a)
Question 1.
Write down the order of each matrix given below and the number of elements in each :
(i) \(\left[\begin{array}{cc}
1 & -1 \\
7 & 4
\end{array}\right]\)
(ii) \([\sin \theta \cos \theta]\)
(iii) \(\{2 4 6 8\}\)
(iv) \(\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right]\)
(v) \(\left[\begin{array}{llll}
2 & 0 & 0 & 0 \\
0 & 2 & 0 & 0 \\
0 & 0 & 2 & 0
\end{array}\right]\)
(vi) \(\left[\begin{array}{rrrrr}
1 & -5 & 0 & 8 & 4 \\
2 & -7 & 3 & 5 & 2 \\
0 & -2 & 1 & 4 & 9
\end{array}\right]\)
(vii) \(\left[\begin{array}{cc}
1 & -2 \\
3 & -5 \\
5 & -9 \\
7 & 0
\end{array}\right]\)
Solution:
Order of matrix is given below :
Order of matrix is given below :
(i) 2 x 2; 4
(ii) 1 x 2; 2
(iii) 1 x 4; 4
(iv) 3 x 1; 3
(v) 3 x 4; 12
(vi) 3 x 5; 15
(vii) 4 x 2; 8
Question 2.
Classify the following matrices as equal or not equal.
(i) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right] ;\left[\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right]\)
(ii) \(\left[\begin{array}{ll}
4 & 7 \\
3 & 2
\end{array}\right] ;\left[\begin{array}{cc}
3+1 & \sqrt{49} \\
5-2 & \frac{6}{3}
\end{array}\right]\)
Solution:
(i) Not equal as their corresponding element are not same.
(ii) \(\left[\begin{array}{ll}
4 & 7 \\
3 & 2
\end{array}\right] \text { and }\left[\begin{array}{cc}
3+1 & \sqrt{49} \\
5-2 & \frac{6}{3}
\end{array}\right]=\left[\begin{array}{ll}
4 & 7 \\
3 & 2
\end{array}\right]\)
These matrices are equal as their corresponding elements are same.
Question 3.
Construct 2 x 2 matrix A where elements aaj are give by a = (2i – j)².
Solution:
Let the matrix 2 x 2 be A\(\left[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right]\)
and it is given that aij = (2i – j)²
∴ a11 = (2 x 1 – 1)² = (2 – 1) = (1)² = 1
a12 = (2 x 1 – 2)² = (2 – 2)² = 0
a21 = (2 x 2 – 1)² = (4 – 1)² = (3)² = 9
a22 = (2 x 2 – 2)² = (4 – 2)² = (2)² = 4
∴ Matrix A = \(\left[\begin{array}{ll}
1 & 0 \\
9 & 4
\end{array}\right]\)
Question 4.
For the matrix B = \(\left[\begin{array}{cc}
3 & 11 \\
-5 & 6 \\
8 & 0
\end{array}\right]\),
(i) What is the order of matrix B ?
(ii) State the elements, a12, a31, a22.
(iii) B is a square matrix. True or False.
Solution:
Matrix B = \(\left[\begin{array}{cc}
3 & 11 \\
-5 & 6 \\
8 & 0
\end{array}\right]\)
(i) Order of matrix is 3 x 2
(ii) Elements a12 = 11, a31 = 8, a22 = 6
(iii) Matrix B is not a square matrix as in it. no. of rows ≠ no. of columns
Question 5.
(a) A matrix has 6 elements. Write the possible orders of the matrix.
(b) If a matrix has 3 rows and 4 columns, what is the number of elements in the matrix ?
Solution:
(a) A matrix which has 6 elements can be of the following possible orders.
6 x 1, 3 x 2, 2 x 3, 1 x 6
(b) A matrix which has 3 rows and 4 columns will have 3 x 4 = 12 elements.
Question 6.
Find x and y such that
(i) \(\left[\begin{array}{cc}
x & y \\
-1 & 5
\end{array}\right]=\left[\begin{array}{ll}
-2 & 0 \\
-1 & 5
\end{array}\right]\)
(ii) \(\left[\begin{array}{ll}
x & 4
\end{array}\right]=\left[\begin{array}{ll}
-2 & y
\end{array}\right]\)
(iii) \(\left[\begin{array}{cc}
2 x & 3 \\
0 & y-1
\end{array}\right]=\left[\begin{array}{cc}
x-4 & 3 \\
0 & 5
\end{array}\right]\)
Solution:
(i) \(\left[\begin{array}{cc}
x & y \\
-1 & 5
\end{array}\right]=\left[\begin{array}{ll}
-2 & 0 \\
-1 & 5
\end{array}\right]\)
Comparing their cooresponding element, we get
x = – 2, y = 0
(ii) \(\left[\begin{array}{ll}
x & 4
\end{array}\right]=\left[\begin{array}{ll}
-2 & y
\end{array}\right]\)
Comparing their corresponding elements, we get
x = – 2, y = 4
(iii) \(\left[\begin{array}{cc}
2 x & 3 \\
0 & y-1
\end{array}\right]=\left[\begin{array}{cc}
x-4 & 3 \\
0 & 5
\end{array}\right]\)
Comparing their corresponding elements, we get
and 2x = x – 4
⇒ 2x – x = – 4
⇒ x = – 4 and y – 1 = 5
⇒ y = 5 + 1 = 6
∴ x = – 4, y = 6
Question 7.
Find p, q, r and s, if \(\left[\begin{array}{ll}
p+4 & 2 q-7 \\
s-3 & r+2 s
\end{array}\right]\) = \(\left[\begin{array}{cc}
6 & -3 \\
2 & 14
\end{array}\right]\).
Solution:
\(\left[\begin{array}{cc}
p+4 & 2 q-7 \\
s-3 & r+2 s
\end{array}\right]=\left[\begin{array}{cc}
6 & -3 \\
2 & 14
\end{array}\right]\)
Comparing their corresponding elements, we get
p + 4 = 6 ⇒ p = 6 – 4 = 2
2q – 7 = – 3 ⇒ 2q = – 3 + 7 ⇒ 2q = 4
⇒ q = \(\frac { 4 }{ 2 }\) = 2
s – 3 = 2 ⇒ s = 2 + 3 = 5
r + 2s = 14 ⇒ r + 2 x 5 = 14 ⇒ r + 10 = 14
⇒ r = 14 – 10 = 4
Hence p = 2, q = 2, r = 4, s = 5
Question 8.
Answer true or false :
(i) Every zero matrix is a square matrix.
(ii) A unit matrix is a diagonal matrix.
(iii) A zero matrix of order 3 is a diagonal matrix.
(iv) \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\) is a unit matrix.
(v) A matrix is an aggregate of numbers.
(vi) \(\left[\begin{array}{ll}
3 & x \\
0 & 2
\end{array}\right]=\left[\begin{array}{ll}
3 & 4 \\
1 & 2
\end{array}\right]\), if x = 4,
(vii) \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\) is the identity matrix for addition of 2 x 2 matrix.
Solution:
(i) False : It is not compulsary.
(ii) True : In unit matrix, diagonal elements are unity and others are zero.
(iii) False : In zero matrix, all element are zero but in diagonal matrix, it is not necessary.
(iv) False : In unit matrix, only diagonal elements are unity.
(v) False : A matrix is an array of real numbers arranged in rows and columns.
(vi) False : Their elements a21 ≠ b21 (0 ≠ 1)
(vii) True.
Question 9.
The length, width and height of two boxes are 6,5,10 and 5, 2, 8 respectively. Write this set of numbers in matrix form so that the first column indicates length, the second indicates width, and the third indicates height. A third box has corresponding dimensions of 7, 3, 3. Write a three by three matrix describing the dimensions of the three boxes.
Solution:
Length, width and height of one box are 6, 5, 10
and that of second box are 5, 2, 8
∴ and of third box are 7, 3, 3
Now we shall write them in Matrix of 3 x 3
order = \(\left[\begin{array}{ccc}
6 & 5 & 10 \\
5 & 2 & 8 \\
7 & 3 & 3
\end{array}\right]\)
Question 10.
(i) Three pupils in an algebra class score marks in three tests as follows : Akhil, 79, 87, 92; Rajnish, 95, 98, 91; Sudha, 76, 88, 77.
Display this information as a 3 x 3 matrix.
(ii) If the price of a record is Rs. 30, of a blade packet Rs. 1.50 and of a soap cake Rs. 1.70, display the prices as a 3 x 1 price matrix.
Solution:
(i) Three pupils Akhil, Rajnish and Sudha got marks in three tests algebra. Their scores are
Akhil = 79, 87, 92; Rajnish = 95, 98, 91 and Sudha = 76, 88, 77
We can display this information in a matrix of 3 x 3 order:
\(\left[\begin{array}{lll}
79 & 87 & 92 \\
95 & 98 & 91 \\
76 & 88 & 77
\end{array}\right]\)
(ii) The price of a record is Rs. 30, of a blade packet is Rs. 1.50 and a soap cake is Rs. 1.70.
We can display them in a matrix of 3 x 1, as
given \(\left[\begin{array}{c}
30 \\
1.50 \\
1.70
\end{array}\right]\)