Using properties of determinates prove ...

CBSE Class 12 Math 2020 Outside Delhi Set 1 Solved Paper

Section - D

Q. Nos. 33 to 36 carry 6 marks each.

Question : 33 of 36

Marks: +1, -0

Using properties of determinates prove that:
|

a−b	b+c	a
b−c	c+a	b
c−a	a+b	c

|=a3+b3+c3−3abc
OR
If A=[

1	3	2
2	0	−1
1	2	3

], then show that A3−4A2−3A+11I=O. Hence find A−1

Solution: 👈: Video Solution

a−b	b+c	a
b−c	c+a	b
c−a	a+b	c

|
=|

−b	b+c+a	a
−c	c+a+b	b
−a	a+b+c	c

|[‌ Applying ‌C1→C1−C3‌ and ‌C2→C2+C3]
=(−1)(a+b+c)|

b	1	a
c	1	b
a	1	c

|
[Taking (−1) common from C1 and (a+b+c) common from C2 ]
=(−1)(a+b+c)|

b	1	a
c−b	0	b−a
a−b	0	c−a

|[‌ Applying ‌R2→R2−R1‌ and ‌R3→R3−R1]
=(−1)(a+b+c)[−(c−b)(c−a)+(b−a)(a−b)]
=(−1)(a+b+c)[−c2+ac+bc−ab+ba−b2−a2+ab]
=(−1)(a+b+c)(−a2−b2−c2+ab+bc+ac)
=(a+b+c)(a2+b2+c2−ab−bc−ac)
=a3+ab2+ac2−a2b−abc−a2c+ba2+b3+bc2−ab2−b2c−abc+ca2+c
b2+c3−acb−bc2−ac2
=a3+b3+c3−3abc

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