Class 12 RD Sharma Solutions β Chapter 6 Determinants β Exercise 6.5
Question 1. Solve each of the following system of homogeneous linear equations:
x + y β 2z = 0
2x + y β 3z =0
5x + 4y β 9z = 0
Solution:
Given:
x + y β 2z = 0
2x + y β 3z =0
5x + 4y β 9z = 0
This system of equations can be expressed in the form of a matrix AX = B
Now find the determinant,
= 1(1 Γ (-9) β 4 Γ (-3)) β 1(2 Γ (-9) β 5 Γ (-3)) β 2(4 Γ 2 β 5 Γ 1)
= 1(-9 + 12) β 1(-18 + 15) β 2(8 β 5)
= 1 Γ 3 β 1 Γ (-3) β 2 Γ 3
= 3 + 3 β 6
= 0
So, D = 0, that means this system of equations has infinite solution.
Now,
Let z = k
β x + y = 2k
And 2x + y = 3k
Now using the Cramerβs rule
x =
x =
x =
x = k
Similarly,
y =
y =
y =
y = k
Therefore,
x = y = z = k.
Question 2. Solve each of the following system of homogeneous linear equations:
2x + 3y + 4z = 0
x + y + z = 0
2x + 5y β 2z = 0
Solution:
2x + 3y + 4z = 0
x + y + z = 0
2x + 5y β 2z = 0
This system of equations can be expressed in the form of a matrix AX = B
Find the determinant
= 2(1 Γ (-2) β 1 Γ 5) β 3(1 Γ (-2) β 2 Γ 1) + 4(1 Γ 5 β 2 Γ 1)
= 2(-2 β 5) β 3(-2 β 2) + 4(5 β 2)
= 2 Γ (-7) β 3 Γ (-4) + 4 Γ 3
= -14 + 12 + 12
= -10
Hence, D β 0, so the system of equation has trivial solution.
Therefore, the system of equation has only solution as x = y = z = 0.
Question 3. Solve each of the following system of homogeneous linear equations:
3x + y + z = 0
x β 4y + 3z = 0
2x +5y β 2z = 0
Solution:
Given:
3x + y + z = 0
x β 4y + 3z = 0
2x +5y β 2z = 0
This system of equations can be expressed in the form of a matrix AX = B
Find the determinant
= 3(8 β 15) β 1(-2 β 6) + 1(13)
= -21 + 8 + 13
= 0
So, the system has infinite solutions:
Let z = k,
So,
3x + y = -k
x β 4y = -3k
Now,
x =
y =
x =
y =
z = k
and there values satisfy equation 3
Hence, x = -7k, y = 8k, z = 13k
Question 4. Find the real values of Ξ» for which the following system of linear equations has non-trivial solutions
2Ξ»x β 2y + 3z = 0
x + Ξ»y + 2z = 0
2x + Ξ»z = 0
Solution:
Finding the determinant
= 3Ξ»3 + 2Ξ» β 8 β 6Ξ»
= 2Ξ»3 β 4Ξ» β 8
Which is satisfied by Ξ» = 2 {for non-trivial solutions Ξ» =2}
Now let z = k
4x β 2y = -3k
x + 2y = -3k
x =
y =
Hence, the solution is x = -k, y = , z = k
Question 5. If a, b, c are non-zero real numbers and if the system of equations
(a β 1)x = y + z
(b β 1)y = z + x
(c β 1)z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc
Solution:
Finding the determinant
Now for non-trivial solution, D = 0
0 = (a β 1)[(b β 1)(c β 1) β 1]+1[-c + 1 β 1] + [-c + 1 β 1] β [ 1 + b β 1]
0 = (a β 1)[bc β b β c + 1 β 1] β c β b
0 = abc β ab -ac + b + c β c β b
ab + bc + ac = abc
Hence proved