Question:
Consider the following statements in respect of the determinant D = kk k kk ()++ ++ 22 11 21 21 33 1 I. D is positive if k > 0. II. D is negative if k < 0. III. D is zero if k = 0. How many of the statements given above are correct?
Consider the following statements in respect of the determinant D = kk k kk ()++ ++ 22 11 21 21 33 1 I. D is positive if k > 0. II. D is negative if k < 0. III. D is zero if k = 0. How many of the statements given above are correct?
Options:
- a) None
- b) One
- c) T wo
- d) All three
- a) None
- b) One
- c) T wo
- d) All three
correct answer : a)
Explanation:
D =
kk k
kk
2
22 11
21 21
33 1
++
++
= (k2 + 2k)(k + 2 -3) – (2k + 1)(2k + 1 – 3) + (6k
+ 3 – 3k – 6)
= k3 – k2 + 2k2 – 2k – 4k2 + 4k – 2k + 2 + 3k – 3
= k3 – 3k2 + 3k – 1
= (k – 1)3
(i) (k – 1)3 > 0 Þ k > 1
(ii) (k – 1)3 < 0 Þ k < 1
(iii) (k - 1)3 =0 Þ k = 1
D =
kk k
kk
2
22 11
21 21
33 1
++
++
= (k2 + 2k)(k + 2 -3) – (2k + 1)(2k + 1 – 3) + (6k
+ 3 – 3k – 6)
= k3 – k2 + 2k2 – 2k – 4k2 + 4k – 2k + 2 + 3k – 3
= k3 – 3k2 + 3k – 1
= (k – 1)3
(i) (k – 1)3 > 0 Þ k > 1
(ii) (k – 1)3 < 0 Þ k < 1
(iii) (k - 1)3 =0 Þ k = 1