Solutions 4. Determinants - Exercise 4.6 | Class 12 Mathematics-I - Toppers Study

Exercise - 4.6

Examine the consistency of the system of equations in Exercises 1 to 6.

Ques.1.  

Ans. Matrix form of given equations is AX = B

 

  A =  and B = 

  = 

There has Unique solution and hence equations are consistent.

 

Ques.2.  

Ans. Matrix form of given equations is AX = B

  

  A =  and B = 

  = 

There has Unique solution and hence equations are consistent.

 

 

Ques.3. 

Ans. Matrix form of given equations is AX = B

  

 A =  and B = 

  = 6 – 6 = 0

Now (adj. A) B =  = 

Therefore, given equations are inconsistent, i.e., have no common solution.

 

 

Ques.4.  

Ans. Matrix form of given equations is AX = B

 

A = 

 

  =   0

There has Unique solution and hence equations are consistent.

 

 

Ques.5.  

Ans. Matrix form of given equations is AX = B

  

 A = 

 

  =  = 

Now (adj. A) = 

And (adj. A) B =  =  = 

hance, given equations are inconsistent.

 

 

Ques.6.  

Ans. Matrix form of given equations is AX = B

  

 A = 

 

  = 

There has Unique solution and hence equations are consistent.

 

 

Ques.7.  

Ans. Matrix form of given equations is AX = B

  

  A = , X =  and B = 

   = 

so, solution is unique and  = 

 

= 

hance,  and 

 

 

Ques.8.  

Ans. Matrix form of given equations is AX = B

 

A = , X =  and B = 

  = 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.9.  

Ans. Matrix form of given equations is AX = B

 

 A = , X =  and B = 

  = 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.10.  

Ans. Matrix form of given equations is AX = B

 

  A = , X =  and B = 

  = 

so, solution is unique and  = 

  = 

hance,  and 

 

Ques.11.  

Ans. Matrix form of given equations is AX = B

 

 A =  , X =  and B = 

  =  = 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.12.  

Ans. Matrix form of given equations is AX = B

 

  A =  , X =   and B = 

 

= 

= 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.13.  

Ans. Matrix form of given equations is AX = B

 

 A =  , X =   and B = 

 

= 

= 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.14.  

Ans. Matrix form of given equations is AX = B

 

 A =  , X =   and B =  

 

= 

= 

so, solution is unique and  = 

 

= 

= 

hance,  and 

 

 

Ques.15. If A =  find  Using  solve the system of equations    

Ans. Given: Matrix A = 

 

  = 

  ……….(i)

Now,  and  and 

 adj. A =  = 

 From eq. (i),

= 

Now, Matrix form of given equations is AX = B

 

Here A =  , X =   and B = 

so, solution is unique and 

 

= 

= 

hance,  and 

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

Ans. Let ` ` ` per kg be the prices of onion, wheat and rice respectively.

 According to given data, we have three equations,

Matrix form of given equations is AX = B

 

 A =  , X =   and B = 

  =  = 

so, solution is unique and  =  …….(i)

Now,   

 adj. A = 

 From eq. (i),

= 

= 

so,  and 

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