Class 12 Maths NCERT Solutions for Chapter 4 Determinants Exercise 4.2
![Class 12 Maths NCERT Solutions for Chapter 4 Determinants Exercise 4.2 Class 12 Maths NCERT Solutions for Chapter 4 Determinants Exercise 4.2](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbcaqHA02UwutwPsFfShn3w6gz4DlQWUBjqKojcWczT7QH8nU-eYlCTpQRjMlUbDgjqM3DhJbUfQluevTHX37MSNccAValPHepJBVd8tHp3jTacIozeHy9Cg9sf_mi6Jd0WaL0RSYKsWncCmG49fiGs9nSAJBBenHsVMRHjlzHJKGqzz2E49kcJj8R/w666-h315-rw/cbse-class-12-maths-ncert-solutions-chapter-4-exercise-4-2.jpg)
Determinants Exercise 4.2 Solutions
1. Using the property of determinants and without expanding, prove that :
Solution
Here, two columns of each determinant are identical.
2. Using the property of determinants and without expanding, prove that :
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoshyfvKRwD28SmCt4qKwI80pgenYRPCbECu3dRKJBFLHRThUn1IRs1XCwWsEGshNGd3eVk72QKHg64ovGtFJWXf6hGAIsMZZzE7iOGaP1ZF8ImpFKH6W2TpyuxXr0VYXiqF2zFIKesBsuFNXq5a944CEVjVxoTOX2vawrJjwsVTsC32q413dgG8lo/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%203.JPG)
Solution
3. Using the property of determinants and without expanding, prove that :
Solution
4. Using the property of determinants and without expanding, prove that :
Solution
5. Using the property of determinants and without expanding, prove that :
Solution
Hence, the given result is proved.
6. By using properties of determinants, show that:
Solution
We have,
Here, the two rows R1 and R3 are identical.
∴ Δ = 0.
7. By using properties of determinants, show that :
Solution
8. By using properties of determinants, show that :
Solution
(i) Let Δ = ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9bAUZcccaQNra6yMcw_hBww3Tv7I41pHAlUeAFWJ1r1XOGQxcNAEjHvGeR0Dzu8ahAP_yDkSmPzr1VVw46TjjMM1KuZTCjgKsliCRJCnF0heZqknXYXqcMp4YhZsHOR6ZLGBm2KbLYz0iI-sXdcItsBZclO7PxlA0-nYk_IpFyI9sX3e3R_pfqSLw/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2016.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIXCe4LwA7IPBBcEJP9T_PA4n0WTXOq9XGZR46Wq6xZL3Q_6vrwxrXqzDBBHOVUgrs-4NM9iDTT14TMlfHib7zPQzpSd67xdJE5VOKC0vIEx0e07tUhoFqPw31EHmngVyGgTo-kKg6JgVDwTThRS0ztBEwR4m9A-NKDQapp30BfHuubwyLFO-2_kku/s320-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2026.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiE3r3NsDtZXvv7joNJWhmHeeyPtABG1iarXYtC4Nqq3fTkpU4n2Q9WNZDQFWyl4xzFr-6JlDs4emQ5OH9VksJuYXJ38cJjm0X5-9oGMLHmhtAvAkiBPAADN0TJAA3c1uUNLD9pWHFAVQtdSpgM1A1iM7a14rV5enH35sPm6RrWV-tCkOtPUJgNRTwF/w329-h633-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2027.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAX9w0YF2ygl9QN9PpfumjKU0z5RphejqQ_L-jvsV96hh1f4zYAwxoOcnWLJIkahKbsfq7RVvYScKsIqYN7lgIxH_UiN12HOTHDfLg1YyL1lx9l_84HP0ZheDBdlu18JuyVoMvlVDCc-xO2yXwAvDFChtAvmBHYGACHfgXmD7dJ9TyErgdq7anpd4Y/w395-h681-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2028.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0vZcNQf9fwoE-6ID8aW9C1eG_CTeB4ag3p8XBboWNqeTEJ4il-05Xr6ISacVYTdTJSkU6_-r4mBni2Iu-K31jTI5SAAQU2W4vyzmHttcUopYxzIN1pjlMJYZlpy8d5CmGp3XttDKeAWGm4Lr07DwoRf6CDiRSI3-mBSfeGKcy24YRibXW8xcxheT4/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2029.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjfK-JHS_kN8OsncKyHQ9FSN7d3_0cAQGzaLk15k5bgjRrCkyegs-hagrAAC3YsgekP2izBjM6Asp3qiYJ9GORxMbGBr14xl3nN6qSDywKMoGZOWUn-N9SDiTkm4m1K2A72irlKLdOewYze6LYy-NJ2BcPnSTEL9jenED35Z5KeWybVd-P7KFmQg1k/w358-h811-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2030.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPEW2Qv4YvZk05RLBcFqma7uAeNNjFsqbMx_bE90IENj99weQCf_e-mXPxhZaTMvLAKhdN8P0o5wxTGpFIZRzATGYI12hJ_I9avLakbnMTGCG8MiRWeoXQxnIgt0QgDjsdDvUt33xtaE_3eB1M_wB2fD2ewGjcxiM43t0zykommVRwnCki2DG4FV-Y/w392-h484-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2032.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr4sDLI6n8nW9Iu8f_D9cagy5bIs-dcPzgmta2PCQj7RkYA7s95mlS1zwgPjAZZP2SVroSIt9hKsqr8RAJD5EOUNdBT9Zz5anAsrkXyU3uDMUgJTB6B6eaU-MMGPux-gFgLjOMvAWNN2g_GhgeSUy6dFcLTZdfemc9JmvqgfMeIoEbjf0y04tDOne0/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2033.JPG)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZxVG2LeYHfEpOinEdSesUQK27M2J38ZBqqi_ksetRwB9hc6IA9FTyswcHpU4uJSOXkHOv1GhxaY_6fPD90uh-MqEPm6o8mDVlZKGs9DR7trAtUsBzemhurw-LgqvNylbN5XJFSGmdPqP7QJzX93qhFbrOa_9ytzjZ5pwdWylqaofjgph5Y79LXf0j/w544-h708-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2034.JPG)
Applying R1 → R1 → R3 and R2 → R2 →R3 , we have :
(ii) Let Δ =
Applying C1 → C1 → C3 and C2 → C2 →C3 , we have :
= (a - b)(b - c)(c - a)(a + b + c)
Hence, the given result is proved.
9. By using properties of determinants, show that :
= (x - y)(y - z)(z - x)(xy + yz + zx)
Solution
Let Δ =
Applying R2 → R2 → R1 and R3 → R3 → R1 , we have :
= (x - y)(z - x)(z- y)[(-xz - yz) + (-x2 – xy + x2 )]
= -(x - y)(z -x)(z - y)(xy + yz + zx)
= (x - y)(y - z)(z - x)(xy + yz + zx)
Hence, the given result is proved.
10. By using properties of determinants, show that :
Solution
(i)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihAYDbYDmBt5f92guQ2RV3kDvddxrtxF-z4AGQ3X-12rcIimjBI1ZtoBL2zLiuVkxwrx8Lopx34D-j79atFWoZyQmQAVJ_3v95TYl2fR65fBJ-eMBZjtPSeqd2Vpy5yYqnndlNoVdDmWz6EecyG_geKdgf9hkmWwIhMQFYlDzITlkY5uqn-_YrnxTn/w327-h570-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2024.JPG)
(ii)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2ictxECt17IAmxFufT_DGaz4JcxK5xM1382g3lIRUtRLLv4eEh8P_JkoSGC5LrfgTnbip9_hhHix4QoJeKkfcrVUcwcVmzzYZ1oVDgsqSKepewd6OcKpRJY3A_aMo9JJIvfjYtju3ZlfOXKjAJUGoJIvo785HuKM87hbrr2moHRwPEkq76fMLJT4M/w345-h516-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2025.JPG)
11. By using properties of determinants, show that :
Solution
12. By using properties of determinants show that :
Solution
13. By using properties of determinants show that :
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYVKwIXiMx-Wq2H6DWOzXUbmjKgpJZzyqDkFZAgbXyiezrzHXXcU1QZDJuRyKsIF5I8W05dftmzl392LMuUqQ24bjfStfYt8Wtp-d8BGtbvnChtr4649C-ixk7TGX_i3pgL68V4eO2qKGy5-3dHNZGMMGVoabmA-vAyIGRKoeXXRi5ktzrAlaEtXuE/s320-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2031.JPG)
Solution
14. By using properties of determinants show that :
Solution
15. Let A be a square matrix of order 3 × 3, then | kA| is equal to
(A) k|A|
(B) k2 | A |
(C) k3 | A |
(D) 3k | A |
(A) k|A|
(B) k2 | A |
(C) k3 | A |
(D) 3k | A |
Solution
Let A = ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil4QIy3pqgmxc0dBMuNVgbNSyK84mPbmzR7ZwgFnZDiUNEnjp5oFSdZjLl9yfEgvFSeEkRFvLpWdlf5LPMvAKc_YUqW1HMVfa9rN4yr_NqZ6XWJLP8eDwV4riZV5wXtBPUZHOap381j7LVBAbC1Z0Lrbjqma-A5X5cTvae0lhcgKFidiXOim-Ng_qX/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2035.JPG)
Then,
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSOASa7mFQjuhL2P6XNiQeyIjcexg8bhD09vTyRO8NGWLMvcCoBQAukV3Gyqs84FS-V_QuwoPzNNX60Ey5a5xdel6gcVGVPtNS1_P9q_HlGwAsJkqFg3agDqc9gKYzlsQxq0bqHQ_n-VIlAB2vCTXE-_6qRD66ny82gSUla57ngxUO-uN2x0OPFKSJ/s1600-rw/NCERT%20Solution%20Class%2012%20Chapter%204%20-%20Determinants%20Exercise%204.2%20img%2036.JPG)
Then,
The correct option is C.
16. Which of the following is correct?
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of the above.
(A) Determinant is a square matrix.
(B) Determinant is a number associated to a matrix.
(C) Determinant is a number associated to a square matrix.
(D) None of the above.
Solution
We know that to every square matrix, A = [aij] of order n , we can associate a number called the determinant of square matrix A, where aij = (i, j)th element of A. Thus, the determinant is a number associated to a square matrix. Hence, the correct option is C.