Advertisements
Advertisements
Question
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
Solution
Let |A| = `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|`
Put a = 0
|A| = `|("b" + "c", - "c", - "b"),("b" - "c", "c", "b"),("c" - "b", "c", "b")|`
= `"bc"|("b" + "c", -1, -1),("b" - "c", 1, 1),("c" - "b", 1, 1)|`
Since two columns identical
= bc × 0 = 0
∴ a – 0 is a factor.
That is, a is a factor.
Put b = 0 in |A|
|A| = `|("b", "a", "a" - "b"),("b", "a", "b" - "a"),(-"b", -"a", "a" + "b")|`
= `"ab" |(1, 1, "a" - "b"),(1, 1, "b" - "a"),(-1, -1, "a" + "b")|`
Since two columns identical
= ab × 0 = 0
∴ c – 0 is a factor.
That is, c is a factor.
The degree of the product of the factors abc is 3.
The degree of the product of leading diagonal elements (b + c)(c + a)(a + b) is 3.
∴ The other factor is the constant factor k.
`|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + "b")|` = kabc
Put a = 1
b = 1
c = 1
`|(1 + 1, 1 - 1, 1 - 1),(1 - 1, 1 + 1, 1 - 1),(1 - 1, 1 - 1, 1 + 1)|` = k × 1 × 1 × 1
`|(2, 0, 0),(0, 2, 0),(0, 0, 2)|` = k
2 × 2 × 2 = 8
⇒ k = 8
∴ `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
APPEARS IN
RELATED QUESTIONS
Prove that `|(1 + "a", 1, 1),(1, 1 + "b", 1),(1, 1, 1 + "c")| = "abc"(1 + 1/"a" + 1/"b" + 1/"c")`
If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax2 + 2bx + c = 0
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
Find the value of `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` if x, y, z ≠ 1
If A is a Square, matrix, and |A| = 2, find the value of |A AT|
Solve that `|(x + "a", "b", "c"),("a", x + "b", "c"),("a", "b", x + "c")|` = 0
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)
Identify the singular and non-singular matrices:
`[(2, -3, 5),(6, 0, 4),(1, 5, -7)]`
Find the value of the product: `|(log_3 64, log_4 3),(log_3 8, log_4 9)| xx |(log_2 3, log_8 3),(log_3 4, log_3 4)|`
Choose the correct alternative:
If A = `[("a", x),(y, "a")]` and if xy = 1, then det(AAT) is equal to
Choose the correct alternative:
The value of x, for which the matrix A = `[("e"^(x - 2), "e"^(7 + x)),("e"^(2 + x), "e"^(2x + 3))]` is singular
Choose the correct alternative:
The value of the determinant of A = `[(0, "a", -"b"),(-"a", 0, "c"),("b", -"c", 0)]` is
Choose the correct alternative:
If A = `|(-1, 2, 4),(3, 1, 0),(-2, 4, 2)|` and B = `|(-2, 4, 2),(6, 2, 0),(-2, 4, 8)|`, then B is given by
The remainder obtained when 1! + 2! + 3! + ......... + 10! is divided by 6 is,
If Δ is the area and 2s the sum of three sides of a triangle, then
If P1, P2, P3 are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then `cosA/P_1 + cosB/P_2 + cosC/P_3` is equal to
What is the value of Δ if, Δ = `|(0, sin alpha, - cos alpha),(-sin alpha, 0, sin beta),(cos alpha, - sin beta, 0)|`
`|("b" + "c", "c", "b"),("c", "c" + "a", "a"),("b", "a", "a" + "b")|` = ______.
Let S = `{((a_11, a_12),(a_21, a_22)): a_(ij) ∈ {0, 1, 2}, a_11 = a_22}`
Then the number of non-singular matrices in the set S is ______.
