Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

   

Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h) will help students to clear their doubts quickly.

Intermediate 1st Year Maths 1A Matrices Solutions Exercise 3(h)

Solve the following systems of equations.
(i) by using Cramer’s rule and matrix inversion method, when the coefficient matrix is non-singular.
(ii) by using the Gauss-Jordan method. Also, determine whether the system has a unique solution or an infinite number of solutions, or no solution, and find the solutions if exist.

Question 1.
5x – 6y + 4z = 15
7x + 4y – 3z = 19
2x + y + 6z = 46
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
5 & -6 & 4 \\
7 & 4 & -3 \\
2 & 1 & 6
\end{array}\right|\)
= 5(24 + 3) + 6(42 + 6) + 4(7 – 8)
= 135 + 288 – 4
= 419
Δ1 = \(\left|\begin{array}{ccc}
15 & -6 & 4 \\
19 & 4 & -3 \\
46 & 1 & 6
\end{array}\right|\)
= 15(24 + 3) + 6(114 + 138) + 4(19 – 184)
= 405 + 1512 – 660
= 1917 – 660
= 1257
Δ2 = \(\left|\begin{array}{ccc}
5 & 15 & 4 \\
7 & 19 & -3 \\
2 & 46 & 6
\end{array}\right|\)
= 5(114 + 138) – 15(42 + 6) + 4(322 – 38)
= 1260 – 720 + 1136
= 1676
Δ3 = \(\left|\begin{array}{ccc}
5 & -6 & 15 \\
7 & 4 & 19 \\
2 & 1 & 46
\end{array}\right|\)
= 5(184 – 19) + 6(322 – 38) + 15(7 – 8)
= 825 + 1704 – 15
= 2529 – 15
= 2514
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q1(i)
Solution is x = 3, y = 4, z = 6.

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q1(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q1(ii).1
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q1(ii).2
Solution is x = 3, y = 4, z = 6

(iii) Gauss-Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q1(iii)
∴ Unique solution exists.
Solution is x = 3, y = 4, z = 6.

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 2.
x + y + z = 1
2x + 2y + 3z = 6
x + 4y + 9z = 3
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 3 \\
1 & 4 & 9
\end{array}\right|\)
= 1(18 – 12) – 1(18 – 3) + 1(8 – 2)
= 6 – 15 + 6
= -3
Δ1 = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
6 & 2 & 3 \\
3 & 4 & 9
\end{array}\right|\)
= 1(18 – 12) – 1(54 – 9) + 1(24 – 6)
= 6 – 45 + 18
= -21
Δ2 = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 6 & 3 \\
1 & 3 & 9
\end{array}\right|\)
= 1(54 – 9) – 1(18 – 3) + 1(6 – 6)
= 45 – 15
= 30
Δ3 = \(\left|\begin{array}{lll}
1 & 1 & 1 \\
2 & 2 & 6 \\
1 & 4 & 3
\end{array}\right|\)
= 1(6 – 24) – 1(6 – 6) + 1(8 – 2)
= -18 – 0 + 6
= -12
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q2(i)
Solution is x = 7, y = -10, z = 4

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q2(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q2(ii).1
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q2(ii).2
∴ Solution is x = 7, y = -10, z = 4

(iii) Gauss-Jordan method:
Augmented matrix is A = \(\left[\begin{array}{llll}
1 & 1 & 1 & 1 \\
2 & 2 & 3 & 6 \\
1 & 4 & 9 & 3
\end{array}\right]\)
R2 → R2 – 2R1, R3 → R3 – R1
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q2(iii)
Unique solution exists.
∴ Solution is x = 7, y = -10, z = 4

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 3.
x – y + 3z = 5
4x + 2y – z = 0
-x + 3y + z = 5
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
1 & -1 & 3 \\
4 & 2 & -1 \\
-1 & 3 & 1
\end{array}\right|\)
= 1(2 + 3) + 1(4 – 1) + 3(12 + 2)
= 5 + 3 + 42
= 50
Δ1 = \(\left|\begin{array}{ccc}
5 & -1 & 3 \\
0 & 2 & -1 \\
5 & 3 & 1
\end{array}\right|\)
= 5(2 + 3) + 1(0 + 5) + 3(0 – 10)
= 25 + 5 – 30
= 0
Δ2 = \(\left|\begin{array}{ccc}
1 & 5 & 3 \\
4 & 0 & -1 \\
-1 & 5 & 1
\end{array}\right|\)
= 1(0 + 5) – 5(4 – 1) + 3(20 – 0)
= 5 – 15 + 60
= 50
Δ3 = \(\left|\begin{array}{ccc}
1 & -1 & 5 \\
4 & 2 & 0 \\
-1 & 3 & 5
\end{array}\right|\)
= 1(10 – 0) + 1(20 – 0) + 5(12 + 2)
= 10 + 20 + 70
= 100
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q3(i)
∴ Solution is x = 0, y = 1, z = 2.

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q3(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q3(ii).1
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q3(ii).2
Solution is x = 0, y = 1, z = 2

(iii) Gauss Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q3(iii)
Unique solution exists.
∴ Solution is x = 0, y = 1, z = 2

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 4.
2x + 6y + 11 = 0
6x + 20y – 6z + 3 = 0
6y – 18z + 1 = 0
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
2 & 6 & 0 \\
6 & 20 & -6 \\
0 & 6 & -18
\end{array}\right|\)
= 2(-360 + 36) – 6(-108 – 0)
= -648 + 648
= 0
∴ Cramer’s rule and matrix inversion method cannot be used.
∵ Δ = 0

(ii) Gauss Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q4(ii)
ρ(A) = 2, ρ(AB) = 3
ρ(A) ≠ ρ(AB)
∴ The given system of equations does not have a solution.

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 5.
2x – y + 3z = 9
x + y + z = 6
x – y + z = 2
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
2 & -1 & 3 \\
1 & 1 & 1 \\
1 & -1 & 1
\end{array}\right|\)
= 2(1 + 1) + 1(1 – 1) + 3(-1 – 1)
= 4 + 0 – 6
= -2
Δ1 = \(\left|\begin{array}{ccc}
9 & -1 & 3 \\
6 & 1 & 1 \\
2 & -1 & 1
\end{array}\right|\)
= 9(1 + 1) + 1(6 – 2) + 3(-6 – 2)
= 18 + 4 – 24
= -2
Δ2 = \(\left|\begin{array}{lll}
2 & 9 & 3 \\
1 & 6 & 1 \\
1 & 2 & 1
\end{array}\right|\)
= 2(6 – 2) – 9(1 – 1) + 3(2 – 6)
= 8 – 0 – 12
= -4
Δ3 = \(\left|\begin{array}{ccc}
2 & -1 & 9 \\
1 & 1 & 6 \\
1 & -1 & 2
\end{array}\right|\)
= 2(2 + 6) + 1(2 – 6) + 9(-1 – 1)
= 16 – 4 – 18
= -6
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q5(i)
Solution is x = 1, y = 2, z = 3.

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q5(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q5(ii).1
Solution is x = 1, y = 2, z = 3.

(iii) Gauss-Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q5(iii)
∴ The given equations have a unique solution.
Solution is x = 1, y = 2, z = 3

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 6.
2x – y + 8z = 13
3x + 4y + 5z = 18
5x – 2y + 7z = 20
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
2 & -1 & 8 \\
3 & 4 & 5 \\
5 & -2 & 7
\end{array}\right|\)
= 2(28 + 10) + 1(21 – 25) + 8(-6 – 20)
= 76 – 4 – 208
= -136
Δ1 = \(\left|\begin{array}{ccc}
13 & -1 & 8 \\
18 & 4 & 5 \\
20 & -2 & 7
\end{array}\right|\)
= 13(28 + 10) + 1(126 – 100) + 8(-36 – 80)
= 494 + 26 – 928
= -408
Δ2 = \(\left|\begin{array}{lll}
2 & 13 & 8 \\
3 & 18 & 5 \\
5 & 20 & 7
\end{array}\right|\)
= 2(126 – 100) – 13(21 – 25) + 8(60 – 90)
= 52 + 52 – 240
= -136
Δ3 = \(\left|\begin{array}{ccc}
2 & -1 & 13 \\
3 & 4 & 18 \\
5 & -2 & 20
\end{array}\right|\)
= 2(80 + 36) + 1(60 – 90) + 13(-6 – 20)
= 232 – 30 – 338
= -136
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q6(i)
∴ Solution is x = 3, y = 1, z = 1

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q6(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q6(ii).1
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q6(ii).2
∴ Solution is x = 3, y = 1, z = 1

(iii) Gauss Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q6(iii)
∴ The given equations have a unique solution and Solution is x = 3, y = 1, z = 1.

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 7.
2x – y + 3z = 8
-x + 2y + z = 4
3x + y – 4z = 0
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
2 & -1 & 3 \\
-1 & 2 & 1 \\
3 & 1 & -4
\end{array}\right|\)
= 2(-8 – 1) + 1(4 – 3) + 3(-1 – 6)
= -18 + 1 – 21
= -38
Δ1 = \(\left|\begin{array}{ccc}
8 & -1 & 3 \\
4 & 2 & 1 \\
0 & 1 & -4
\end{array}\right|\)
= 8(-8 – 1) + 1(-16 – 0) + 3(4 – 0)
= -72 – 16 + 12
= -76
Δ2 = \(\left|\begin{array}{ccc}
2 & 8 & 3 \\
-1 & 4 & 1 \\
3 & 0 & -4
\end{array}\right|\)
= 2(-16 – 0) – 8(4 – 3) + 3(-0 – 12)
= -32 – 8 – 36
= -76
Δ3 = \(\left|\begin{array}{ccc}
2 & -1 & 8 \\
-1 & 2 & 4 \\
3 & 1 & 0
\end{array}\right|\)
= 2(0 – 4) + 1(0 – 12) + 8(-1 – 6)
= -8 – 12 – 56
= -76
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q7(i)
∴ Solution is x = 2, y = 2, z = 2.

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q7(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q7(ii).1
Solution is x = 2, y = 2, z = 2

(iii) Gauss Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q7(iii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q7(iii).1
∴ The given equations have a unique solution and solution is x = 2, y = 2, z = 2.

Inter 1st Year Maths 1A Matrices Solutions Ex 3(h)

Question 8.
x + y + z = 9
2x + 5y + 7z = 52
2x + y – z = 0
Solution:
(i) Cramer’s rule:
Δ = \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
2 & 5 & 7 \\
2 & 1 & -1
\end{array}\right|\)
= 1(-5 – 7) – 1(-2 – 14) + 1(2 – 10)
= -12 + 16 – 8
= -4
Δ1 = \(\left|\begin{array}{ccc}
9 & 1 & 1 \\
52 & 5 & 7 \\
0 & 1 & -1
\end{array}\right|\)
= 9(-5 – 7) – 1(-52 – 0) + 1(52 – 0)
= -108 + 52 + 52
= -4
Δ2 = \(\left|\begin{array}{ccc}
1 & 9 & 1 \\
2 & 52 & 7 \\
2 & 0 & -1
\end{array}\right|\)
= 1(-52 – 0) – 9(-2 – 14) + 1(0 – 104)
= -52 + 144 – 104
= -12
Δ3 = \(\left|\begin{array}{ccc}
1 & 1 & 9 \\
2 & 5 & 52 \\
2 & 1 & 0
\end{array}\right|\)
= 1(0 – 52) – 1(0 – 104) + 9(2 – 10)
= -52 + 104 – 72
= -20
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q8(i)

(ii) Matrix inversion method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q8(ii)
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q8(ii).1
Solution is x = 1, y = 3, z = 5

(iii) Gauss Jordan method:
Inter 1st Year Maths 1A Matrices Solutions Ex 3(h) Q8(iii)
∴ The given equations have a unique solution and solution is x = 1, y = 3, z = 5.