NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.2
Page No 119:
Question 1:
Using the property of determinants and without expanding, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6433/Chapter%204_html_m34ceacad.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6433/Chapter%204_html_m7e16c551.gif)
Question 2:
Using the property of determinants and without expanding, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6434/Chapter%204_html_m54f310dc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6434/Chapter%204_html_63901485.gif)
Here, the two rows R1 and R3 are identical.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6434/Chapter%204_html_4dd19828.gif)
Question 3:
Using the property of determinants and without expanding, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6435/Chapter%204_html_7ba49357.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6435/Chapter%204_html_m7dd04180.gif)
Question 4:
Using the property of determinants and without expanding, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6436/Chapter%204_html_393dc1bf.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6436/Chapter%204_html_7c73293c.gif)
By applying C3 → C3 + C2, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6436/Chapter%204_html_3998e583.gif)
Here, two columns C1 and C3 are proportional.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6436/Chapter%204_html_4dd19828.gif)
Question 5:
Using the property of determinants and without expanding, prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_21fba1a3.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m76aec2fc.gif)
Applying R2 → R2 − R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m1d615a3f.gif)
Applying R1 ↔R3 and R2 ↔R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m1cf7fb93.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m5f94847d.gif)
Applying R1 → R1 − R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m224a7b62.gif)
Applying R1 ↔R2 and R2 ↔R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_5299b3f6.gif)
From (1), (2), and (3), we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6437/Chapter%204_html_m1a942179.gif)
Hence, the given result is proved.
Page No 120:
Question 6:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6438/Chapter%204_html_79eca739.gif)
Answer:
We have,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6438/Chapter%204_html_m29d1e3cc.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6438/Chapter%204_html_3a09dee6.gif)
Here, the two rows R1 and R3 are identical.
∴Δ = 0.
Question 7:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6439/Chapter%204_html_53198e9d.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6439/Chapter%204_html_m1043572.gif)
Applying R2 → R2 + R1 and R3 → R3 + R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6439/Chapter%204_html_6d7f53fb.gif)
Question 8:
By using properties of determinants, show that:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m563fb563.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m563fb563.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m214eb6d4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m214eb6d4.gif)
Answer:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_89506c2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_89506c2.gif)
Applying R1 → R1 − R3 and R2 → R2 − R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_6f69e6e9.gif)
Applying R1 → R1 + R2, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_7cb13fc9.gif)
Expanding along C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m27875559.gif)
Hence, the given result is proved.
(ii) Let
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m2790483d.gif)
Applying C1 → C1 − C3 and C2 → C2 − C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m246c534f.gif)
Applying C1 → C1 + C2, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m3415f1ee.gif)
Expanding along C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6440/Chapter%204_html_m4f62d366.gif)
Hence, the given result is proved.
Question 9:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6442/Chapter%204_html_355485c4.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6442/Chapter%204_html_2c7a9498.gif)
Applying R2 → R2 − R1 and R3 → R3 − R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6442/Chapter%204_html_3b60a4ce.gif)
Applying R3 → R3 + R2, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6442/Chapter%204_html_m333415ed.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6442/Chapter%204_html_2e7e34fa.gif)
Hence, the given result is proved.
Question 10:
By using properties of determinants, show that:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m7404ea78.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m7404ea78.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m6ca778c9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m6ca778c9.gif)
Answer:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m50478e14.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m50478e14.gif)
Applying R1 → R1 + R2 + R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m521a4384.gif)
Applying C2 → C2 − C1, C3 → C3 − C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_7159bae1.gif)
Expanding along C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m47d09c0.gif)
Hence, the given result is proved.
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_10ffeea6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_10ffeea6.gif)
Applying R1 → R1 + R2 + R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_7dd23915.gif)
Applying C2 → C2 − C1 and C3 → C3 − C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_m18416f2f.gif)
Expanding along C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6448/Chapter%204_html_7b7c9f87.gif)
Hence, the given result is proved.
Question 11:
By using properties of determinants, show that:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_3bcf218e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_3bcf218e.gif)
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m6b822a94.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m6b822a94.gif)
Answer:
(i) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m2ad7a9a1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m2ad7a9a1.gif)
Applying R1 → R1 + R2 + R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_100f2562.gif)
Applying C2 → C2 − C1, C3 → C3 − C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_5b75f335.gif)
Expanding along C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m70739894.gif)
Hence, the given result is proved.
(ii) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_5f6da264.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_5f6da264.gif)
Applying C1 → C1 + C2 + C3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_m84f5197.gif)
Applying R2 → R2 − R1 and R3 → R3 − R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_10d6ee16.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6451/Chapter%204_html_6c9dd75f.gif)
Hence, the given result is proved.
Page No 121:
Question 12:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6455/Chapter%204_html_m31a8ec4f.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6455/Chapter%204_html_m5af9956a.gif)
Applying R1 → R1 + R2 + R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6455/Chapter%204_html_m6e2e046a.gif)
Applying C2 → C2 − C1 and C3 → C3 − C1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6455/Chapter%204_html_m3ef95028.gif)
Expanding along R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6455/Chapter%204_html_m31e7ad67.gif)
Hence, the given result is proved.
Question 13:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6458/Chapter%204_html_317e675b.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6458/Chapter%204_html_m38800b43.gif)
Applying R1 → R1 + bR3 and R2 → R2 − aR3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6458/Chapter%204_html_m12ea1c0b.gif)
Expanding along R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6458/Chapter%204_html_m744fa485.gif)
Question 14:
By using properties of determinants, show that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_3a589695.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_m281f621f.gif)
Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_m4723eaeb.gif)
Applying R2 → R2 − R1 and R3 → R3 − R1, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_5197aeba.gif)
Applying C1 → aC1, C2 → bC2, and C3 → cC3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_349958b4.gif)
Expanding along R3, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6463/Chapter%204_html_m3c8bec5f.gif)
Hence, the given result is proved.
Question 15:
Choose the correct answer.
Let A be a square matrix of order 3 × 3, then
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_ec77c01.gif)
A.
B.
C.
D.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_1ae7f03a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_6379dd52.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_3feba8d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_5040712e.gif)
Answer:
Answer: C
A is a square matrix of order 3 × 3.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6474/Chapter%204_html_m50c1e7b5.gif)
Hence, the correct answer is C.
Question 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 these
Answer:
Answer: C
We know that to every square matrix,
of order n. We can associate a number called the determinant of square matrix A, where
element of A.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6479/Chapter%204_html_me4867ea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6479/Chapter%204_html_m5a24269d.gif)
Thus, the determinant is a number associated to a square matrix.
Hence, the correct answer is C.