Advertisements
Advertisements
Question
Show that `|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|` = (a + b + c)(a – b)(b – c)(c – a)
Advertisements
Solution
Let |A| = `|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|`
Put a = b in |A|
|A| = `|("b" + "c", "b", "b"^2),("c" + "b", "b", "b"^2),("b" + "b", "c", "c"^2)|`
|A| = `|("b" + "c", "b", "b"^2),("b" + "c", "b", "b"^2),("b" + "b", "c", "c"^2)|`
Since two rows are idenctical
|A| = 0
Since two rows are idenctical
|A| = 0
∴ a – b is a factor of |A|.
The given determinant is in cyclic symmetric form in a, b and c.
Therefore, b – c and c – a are also factors.
The degree of the product of the factors (a – b)(b – c)(c – a) is 3 and the degree of the product of the leading diagonal elements (b + c) . b . c2 is 4.
Therefore, the other factor is k(a + b + c).
`|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|` = k(a + b + c)(a – b) × (b – c)(c – a)
Put a = 1, b = 2, c = 3 we get
`|(2 +3, 1, 1^2),(3 + 1, 2, 2^2),(1 + 2, 3, 3^2)|` = k(1 + 2 + 3)(1 – 2) × (2 – 3)(3 – 1)
`|(5, 1, 1),(4, 2, 4),(3, 3, 9)|` = k × 6 × –1 × –1 × 2
5(18 – 12) – 1(36 – 12) + 1(12 – 6) = 12k
5 × 6 – 24 + 6 = 12k
30 – 24 + 6 = 12k
12 = 12
⇒ k = 1
∴ `|("b" + "C", "a", "a"^2),("c" + "a", "b", "b"^2),("a" + "b", "c", "c"^2)|` = (a + b + c)(a – b)(b – c)(c – a)
APPEARS IN
RELATED QUESTIONS
Prove that `|("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| = 4"a"^2"b"^2"c"^2`
If a, b, c are pth, qth and rth terms of an A.P, find the value of `|("a", "b", "c"),("p", "q", "r"),(1, 1, 1)|`
If a, b, c, are all positive, and are pth, qth and rth terms of a G.P., show that `|(log"a", "p", 1),(log"b", "q", 1),(log"c", "r", 1)|` = 0
If A and B are square matrices of order 3 such that |A| = –1 and |B| = 3, find the value of |3AB|
Using cofactors of elements of second row, evaluate |A|, where A = `[(5, 3, 8),(2, 0, 1),(1, 2, 3)]`
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
Show that `|(1, 1, 1),(x, y, z),(x^2, y^2, z^2)|` = (x – y)(y – z)(z – x)
If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k
Identify the singular and non-singular matrices:
`[(0, "a" - "b", "k"),("b" - "a", 0, 5),(-"k", -5, 0)]`
Determine the values of a and b so that the following matrices are singular:
B = `[("b" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
Choose the correct alternative:
If A = `[(1, -1),(2, -1)]`, B = `[("a", 1),("b", -1)]` and (A + B)2 = A2 + B2, then the values of a and b are
Choose the correct alternative:
If ⌊.⌋ denotes the greatest integer less than or equal to the real number under consideration and – 1 ≤ x < 0, 0 ≤ y < 1, 1 ≤ z ≤ 2, then the value of the determinant `[([x] + 1, [y], [z]),([x], [y] + 1, [z]),([x], [y], [z] + 1)]`
The remainder obtained when 1! + 2! + 3! + ......... + 10! is divided by 6 is,
If f(x) = `|(cos^2x, cosx.sinx, -sinx),(cosx sinx, sin^2x, cosx),(sinx, -cosx, 0)|`, then for all x
What is the value of Δ if, Δ = `|(0, sin alpha, - cos alpha),(-sin alpha, 0, sin beta),(cos alpha, - sin beta, 0)|`
For f(x)= `ℓn|x + sqrt(x^2 + 1)|`, then the value of`g(x) = (cosx)^((cosecx - 1))` and `h(x) = (e^x - e^-x)/(e^x + e^-x)`, then the value of `|(f(0), f(e), g(π/6)),(f(-e), h(0), h(π)),(g((5π)/6), h(-π), f(f(f(0))))|` is ______.
`|("b" + "c", "c", "b"),("c", "c" + "a", "a"),("b", "a", "a" + "b")|` = ______.
If `|(1 + x, x, x^2),(x, 1 + x, x^2),(x^2, x, 1 + x)|` = ax5 + bx4 + cx3 + dx2 + λx + µ be an identity in x, where a, b, c, d, λ, µ are independent of x. Then the value of λ is ______.
