Class 12 NCERT Solutions - Mathematics Part I - Chapter 4 Determinants - Exercise 4.6 | Set 2

Chapter 4 Determinants   - Exercise 4.6 | Set 1

Question 11. 2x + y + z = 1 

                               x - 2y - z = 3/2

                               3y - 5z = 9 

Solution:

Matrix form of the given equation is AX = B

i.e.

∴ |A| =

∴ Solution is unique.

Now, X = A^-1B = (adj.A)B

Therefore, x=1, y=1/2, z=3/2

Question 12.  x - y + z = 4

                                2x + y - 3z = 0

                                x + y + z = 2

Solution:

Matrix form of the given equation is AX = B

i.e

∴ |A| =

∴ Solution is unique.

Now, X = A^-1B = (adj.A)B

Therefore, x = 2, y = -1, z = 1

Question 13. 2x + 3y +3 z = 5 

                               x - 2y + z = - 4 

                               3x - y - 2z = 3

Solution:

Matrix form of given equation is AX = B

i.e.

∴ |A| =

∴ Solution is unique.

Now, X = A^-1B = (adj.A)B

Therefore, x = 1, y = 2, z = -1

Question 14. x - y + 2z = 7

                               3x + 4y - 5z = - 5

                               2x - y + 3z = 12

Solution:

Matrix form of given equation is AX = B

i.e.

∴ |A| =

∴ Solution is unique.

Now, X = A^-1B = (adj.A)B

Therefore, x = 2, y = 1, z = 3

Question 15. If A=, find A–1. Using A–1 solve the system of equations 

2x - 3y + 5z = 11

3x + 2y - 4z = - 5

x + y - 2z = - 3

Solution:

Given: A=

Now, |A|= 

∴ |A|= 

Means, A^-1 exists.

And A^-1 =(adj.A)......(1)

Now,

∴ adj. A =

From eq. (1),

A^-1=

Now, Matrix form of given equation is AX = B

i.e.

∵ Solution is unique.

∴ X=A^-1B

⇒

Therefore, x = 1, y = 2, z = 3

Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is 60 rupees. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is 90 rupees. The cost of 6 kg onion 2 kg wheat and 3 kg rice is 70 rupees. Find cost of each item per kg by matrix method.

Solution:

Let Rs x, Rs y, Rs z per kg be the prices of onion, wheat and rice respectively.

A.T.Q.

4x+3y+2z=60

2x+4y+6z=90

6x+2y+3z=70

Matrix form of equation is AX = B

where, A=,B=and X=

=> 

Now, |A|=

∴ Solution is unique

Now, X=A^-1B=(adj. A)B......(1)

Now, 

∴ (adj.A)= 

From eqⁿ.(1)

Therefore, x = 5, y = 8, z = 8

Hence, the cost of onion, wheat and rice are Rs. 5, Rs 8 and Rs 8 per kg.