NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q1: Write Minors and Cofactors of the elements of following determinants:

Ans: (i) The given determinant is
Minor of element a_ij is M_ij.



Q2: 

Ans:







Q3: Using Cofactors of elements of second row, evaluate Δ =.
Ans: The given determinant is
 .
We have:



We know that ∆ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
∆ = a₂₁A₂₁ + a₂₂A₂₂ + a₂₃A₂₃ = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7.

Q4: Using Cofactors of elements of third column, evaluate Δ=  

Ans: The given determinant is  .

We know that Δ is equal to the sum of the product of the elements of the second row
with their corresponding cofactors.

Hence,

Q5: If ∆ = and A_ij is Cofactors of a_ij , then value of ∆ is given by
(A) a₁₁ A₃₁+ a₁₂ A₃₂ + a₁₃ A₃₃ 
(B) a₁₁ A₁₁ + a₁₂ A₂₁ + a₁₃ A₃₁ 
(C) a₂₁ A₁₁ + a₂₂ A₁₂ + a₂₃ A₁₃ 
(D) a₁₁ A₁₁+ a₂₁ A₂₁ + a₃₁ A₃₁
Ans: D

Given : Δ =
Δ = Sum of products of elements of row (or column) with their corresponding cofactors.
Δ=a₁₁ A₁₁+ a₂₁ A₂₁ + a₃₁ A₃₁
So, option D is correct.

Old NCERT Questions

Hence,

Q2: Examine the consistency of the system of equations.
 x + 2y = 2
 2x + 3y = 3
Ans:
The given system of equations is:
x + 2y = 2
2x + 3y = 3
The given system of equations can be written in the form of AX = B, where

 A is non-singular.
Therefore, A⁻¹ exists.
Hence, the given system of equations is consistent.

Q3: Examine the consistency of the system of equations.
 2x − y = 5 x
 + y = 4
Ans: The given system of equations is:
2x − y = 5 x
+ y = 4
The given system of equations can be written in the form of AX = B, where

 A is non-singular.
Therefore, A⁻¹ exists.
Hence, the given system of equations is consistent.

Q4: Examine the consistency of the system of equations.
 x + 3y = 5
 2x + 6y = 8
Ans: The given system of equations is:
x + 3y = 5
2x + 6y = 8
The given system of equations can be written in the form of AX = B, where

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

Q5: Examine the consistency of the system of equations.
 x + y + z = 1 
2x + 3y + 2z = 2 
ax + ay + 2az = 4
Ans: The given system of equations is:
x + y + z = 1 2x
+ 3y + 2z = 2 ax
+ ay + 2az = 4
This system of equations can be written in the form AX = B, where

 A is non-singular.
Therefore, A⁻¹ exists.
Hence, the given system of equations is consistent.

Q6: Examine the consistency of the system of equations.
 3x − y − 2z = 2
 2y − z = −1
 3x − 5y = 3
Ans: The given system of equations is:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
This system of equations can be written in the form of AX = B, where

A is a singular matrix.



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

Q7: Examine the consistency of the system of equations.
 5x − y + 4z = 5
 2x + 3y + 5z = 2
 5x − 2y + 6z = −1
Ans: The given system of equations is:
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
This system of equations can be written in the form of AX = B, where

Therefore, A⁻¹ exists.
Hence, the given system of equations is consistent.

Q8: Solve system of linear equations, using matrix method.

Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q9: Solve system of linear equations, using matrix method.

Ans:The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q10: Solve system of linear equations, using matrix method.

Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.





Q11: Solve system of linear equations, using matrix method.
 5x + 2y = 3
 3x + 2y = 5
Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.

Q12: Solve system of linear equations, using matrix method.

Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q13: Solve system of linear equations, using matrix method.
 x − y + z = 4 
2x + y − 3z = 0 
x + y + z = 2
Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q14: Solve system of linear equations, using matrix method.
 2x + 3y + 3z = 5
 x − 2y + z = −4
 3x − y − 2z = 3
Ans: The given system of equations can be written in the form AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q15: Solve system of linear equations, using matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Ans: The given system of equations can be written in the form of AX = B, where

Thus, A is non-singular. Therefore, its inverse exists.


Q16: If   find A⁻¹. Using A⁻¹ solve the system of equations
Ans:

Now, the given system of equations can be written in the form of AX = B, where



Q17: The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.
Ans: Let the cost of onions, wheat, and rice per kg be Rs x, Rs y,and Rs z respectively.
Then, the given situation can be represented by a system of equations as:

This system of equations can be written in the form of AX = B, where

Now,
X = A⁻¹ B

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