# EXERCISE 4.6 ( Determinants )

Question 1: Examine the consistency of the system of equations:(Class 12 ncert solution math exercise 4.6)

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

=3-4

So, is non-singular.

Therefore, exists.

Thus, the given system of equations is consistent.

Question 2: Examine the consistency of the system of equations:

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

=2+1

So, is non-singular.

Therefore, exists.

Hence, the given system of equations is consistent.

Question 3: Examine the consistency of the system of equations:

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

=6-6
=0

So, is a singular matrix.

Now,

Therefore,

Thus, the solution of the given system of equations does not exist.

Hence, the system of equations is inconsistent.

Question 4: Examine the consistency of the system of equations:

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists.

Thus, the given system of equations is consistent.

Question 5:Examine the consistency of the system of equations:

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is a singular matrix.

Now,

Therefore,

Thus, the solution of the given system of equations does not exist.

Hence, the system of equations is inconsistent.

Question 6:Examine the consistency of the system of equations:

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is nonsingular. Therefore, exists.

Hence, the given system of equations is consistent.

Question 7: Solve system of linear equations, using matrix method.

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists.

Now,

Then,

Hence, and

Question 8: Solve system of linear equations, using matrix method.

Solution:

The given system of equations is:

The given system of equations can be written in the form of

,
where and

Hence,

So, is non-singular.

Therefore, exists.

Now,

Therefore,

Hence, and

Question 9: Solve system of linear equations, using matrix method.

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is nonsingular. Therefore, exists.

Now,

Therefore,

Hence, and

Question 10: Solve system of linear equations, using matrix method.

Solution:

The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists.

Now,

Therefore,

Hence, and

Question 11: Solve system of linear equations, using matrix method.

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists.

Hence,

Therefore,

Hence, and

Question 12: Solve system of linear equations, using matrix method.

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is nonsingular.

Therefore, exists.

Now,

Hence,

Therefore,

Hence, and

Question 13: Solve system of linear equations, using matrix method.

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists.

Now,

Hence,

Therefore,

.

Hence, and

Question 14:Solve system of linear equations, using matrix method.

Solution:The given system of equations is:

The given system of equations can be written in the form of , where

Hence,

So, is non-singular.

Therefore, exists

Hence,

Therefore,

Hence, and

Question 15:

Solution: It is given that

Therefore,

Now,

Hence,

The given system of equations can be written in the form of , where

The solution of the system of equations is given by .

Therefore,

Hence, and

Question 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 6 kg rice is ₹ 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is ₹ 70 . Find cost of each item per kg by matrix method.

Solution: Let the cost of onions, wheat, and rice per in be and respectively.

Then, the given situation can be represented by a system of equations as:

The given system of equations can be written in the form of , where

Therefore,

So, is non-singular.

Therefore, exists.

Now,

Therefore,

Hence,

Thus, and

Hence, the cost of onions is ₹ 5 per kg the cost of wheat is ₹ 8 per kg, and the cost of rice is ₹ 8 per kg.