Advertisements
Advertisements
प्रश्न
If a + b + c ≠ 0 and `|("a", "b","c"),("b", "c", "a"),("c", "a", "b")|` 0, then prove that a = b = c.
उत्तर
Let Δ = `|("a", "b","c"),("b", "c", "a"),("c", "a", "b")|`
[Applying R1 → R1 + R2 + R3]
Δ = `|("a" + "b" + "c", "a" + "b" + "c", "a" + "b" + "c"),("b", "c", "a"),("c", "a", "b")|`
= `("a"+ "b" + "c")|(1, 1, 1),("b", "c", "a"),("c", "a", "b")|`
[Applying C1 → C1 + C3 and C2 → C2 – C3]
Δ = `("a" + "b" + "c")|(0, 0,1),("b" - "a", "c" - "a", "a"),("c" - "b", "a" - "b", "b")|`
[Expanding along R1]
= `("a" + "b" + "c")[1("b" - "a")("a" - "b") - ("c" - "a")("c" - "b")`
= `("a" + "b" + "c")("ba" - "b"^2- "a"^2 + "ab" - "c"^2 + "cb" + "ac" - "ab")`
= `-("a" + "b" + "c")("a"^2 + "b"^2 + "c"^2 - "ab" - "bc" - "ca")`
= `(-1)/2 ("a" + "b" + "c")[2"a"^2 + 2"b"^2 + 2"c"^2 - 2"ab" - 2"bc" - 2"ca"]`
= `-1/2 ("a" + "b" + "c")[("a"^2 + "b"^2 - 2"ab") + ("b"^2 + "c"^2 - 2"bc") + ("c"^2 + "a"^2 - 2"ac")]`
= `(-1)/2 ("a" + "b" + "c")[("a" - "b")^2 + ("b" - "c")^2 + ("c" - "a")^2]`
Given, Δ = 0
⇒ `(-1)/2 ("a" + "b" + "c")[("a" - "b")^2 + ("b" - "c")^2 + ("c" - "a")^2]` = 0
⇒ `("a" - "b")^2 + ("b" - "c")^2 + ("c" - "a")^2` = 0 ...[∵ a + b + c ≠ 0, given]
⇒ a – b = b – c = c – a = 0
⇒ a = b = c
APPEARS IN
संबंधित प्रश्न
Using the property of determinants and without expanding, prove that:
`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`
Without expanding at any stage, find the value of:
`|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|`
A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?
Let A = [aij] be a square matrix of order 3 × 3 and Cij denote cofactor of aij in A. If |A| = 5, write the value of a31 C31 + a32 C32 a33 C33.
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.
If A is a matrix of order 3 and |A| = 8, then |adj A| = __________ .
Using matrices, solve the following system of linear equations :
x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11
The determinant ∆ = `|(sqrt(23) + sqrt(3), sqrt(5), sqrt(5)),(sqrt(15) + sqrt(46), 5, sqrt(10)),(3 + sqrt(115), sqrt(15), 5)|` is equal to ______.
The value of the determinant ∆ = `|(sin^2 23^circ, sin^2 67^circ, cos180^circ),(-sin^2 67^circ, -sin^2 23^circ, cos^2 180^circ),(cos180^circ, sin^2 23^circ, sin^2 67^circ)|` = ______.
If a1, a2, a3, ..., ar are in G.P., then prove that the determinant `|("a"_("r" + 1), "a"_("r" + 5), "a"_("r" + 9)),("a"_("r" + 7), "a"_("r" + 11), "a"_("r" + 15)),("a"_("r" + 11), "a"_("r" + 17), "a"_("r" + 21))|` is independent of r.
If x + y + z = 0, prove that `|(x"a", y"b", z"c"),(y"c", z"a", x"b"),(z"b", x"c", y"a")| = xyz|("a", "b", "c"),("c", "a", "b"),("b", "c", "a")|`
If f(x) = `|(0, x - "a", x - "b"),(x + "b", 0, x - "c"),(x + "b", x + "c", 0)|`, then ______.
If A = `[(2, lambda, -3),(0, 2, 5),(1, 1, 3)]`, then A–1 exists if ______.
If A is invertible matrix of order 3 × 3, then |A–1| ______.
If A is a matrix of order 3 × 3, then (A2)–1 = ______.
If f(x) = `|((1 + x)^17, (1 + x)^19, (1 + x)^23),((1 + x)^23, (1 + x)^29, (1 + x)^34),((1 +x)^41, (1 +x)^43, (1 + x)^47)|` = A + Bx + Cx2 + ..., then A = ______.
If A and B are matrices of order 3 and |A| = 5, |B| = 3, then |3AB| = 27 × 5 × 3 = 405.
`"A" = abs ((1/"a", "a"^2, "bc"),(1/"b", "b"^2, "ac"),(1/"c", "c"^2, "ab"))` is equal to ____________.
`abs ((1 + "a", "b", "c"),("a", 1 + "b", "c"),("a", "b", 1 + "c")) =` ____________
If A = `[(1,0,0),(2,"cos x","sin x"),(3,"sin x", "-cos x")],` then det. A is equal to ____________.
If `"abc" ne 0 "and" abs ((1 + "a", 1, 1),(1, 1 + "b", 1),(1,1,1 + "c")) = 0, "then" 1/"a" + 1/"b" + 1/"c" =` ____________.
For positive numbers x, y, z the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 3, log_y z),(log_z x, log_z y, 5)|` is
For positive numbers x, y, z, the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 1, log_y z),(log_z x, log_z y, 1)|` is
