NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.6
Page No 136:
Question 1:
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6584/Chapter%204_html_30b0a9c2.gif)
∴ A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.
Question 2:
Examine the consistency of the system of equations.
2x − y = 5
x + y = 4
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6589/Chapter%204_html_625c2816.gif)
∴ A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.
Question 3:
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6594/Chapter%204_html_2c1ee03a.gif)
∴ A is a singular matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6594/Chapter%204_html_10044d87.gif)
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.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6601/Chapter%204_html_m4b4d1993.gif)
∴ A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.
Question 5:
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6606/Chapter%204_html_m54650ebf.gif)
∴ A is a singular matrix.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6606/Chapter%204_html_m1e5b7c2a.gif)
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.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
Answer:
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
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6608/Chapter%204_html_m631ef4b3.gif)
∴ A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.
Question 7:
Solve system of linear equations, using matrix method.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_m724cf69e.gif)
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_m33c86301.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_39a6a942.gif)
Question 8:
Solve system of linear equations, using matrix method.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_1d13fa43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_m2be58a95.gif)
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_59b8eaa8.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_m72cd731.gif)
Question 9:
Solve system of linear equations, using matrix method.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_m326506ee.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_m5e3784fb.gif)
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_5b6948cf.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_324d0cda.gif)
Question 10:
Solve system of linear equations, using matrix method.
5x + 2y = 3
3x + 2y = 5
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6632/Chapter%204_html_m2810aa19.gif)
Thus, A is non-singular. Therefore, its inverse exists.
Question 11:
Solve system of linear equations, using matrix method.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_54e2ee67.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_mcb0a148.gif)
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_m4ae283eb.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_m71cac2ce.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_bf8a970.gif)
Question 12:
Solve system of linear equations, using matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_fffddc7.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_59c7d826.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_2398fa92.gif)
Question 13:
Solve system of linear equations, using matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
Answer:
The given system of equations can be written in the form AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_m6e5aee51.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_2a364aa6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_m480412b3.gif)
Question 14:
Solve system of linear equations, using matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Answer:
The given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_6f61bdbd.gif)
Thus, A is non-singular. Therefore, its inverse exists.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_214d0473.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_m76863815.gif)
Page No 137:
Question 15:
If
, find A−1. Using A−1 solve the system of equations
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_89d4e0a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m4a65d85.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m1d26ecb7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m4a89e4df.gif)
Now, the given system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_1236668a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_24ed4cc2.gif)
Question 16:
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.
Answer:
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:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_25a36f4.gif)
This system of equations can be written in the form of AX = B, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_74fc5d5f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_5c8fdc20.gif)
Now,
X = A−1 B
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_m5d62b9ab.gif)
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.