# Class 12 ncert solution math exercise 4.6

# 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.