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.
Advertisements
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
Find the value of x, if `|(2,4),(5,1)|=|(2x, 4), (6,x)|`.
Find the value of x, if `|(2,3),(4,5)|=|(x,3),(2x,5)|`.
Using the property of determinants and without expanding, prove that:
`|(x, a, x+a),(y,b,y+b),(z,c, z+ c)| = 0`
Use properties of determinants to solve for x:
`|(x+a, b, c),(c, x+b, a),(a,b,x+c)| = 0` and `x != 0`
A matrix A of order 3 × 3 has determinant 5. What is the value of |3A|?
A matrix of order 3 × 3 has determinant 2. What is the value of |A (3I)|, where I is the identity matrix of order 3 × 3.
Which of the following is not correct?
Which of the following is not correct in a given determinant of A, where A = [aij]3×3.
Solve the following system of linear equations using matrix method:
3x + y + z = 1
2x + 2z = 0
5x + y + 2z = 2
Without expanding, show that Δ = `|("cosec"^2theta, cot^2theta, 1),(cot^2theta, "cosec"^2theta, -1),(42, 40, 2)|` = 0
If x = – 4 is a root of Δ = `|(x, 2, 3),(1, x, 1),(3, 2, x)|` = 0, then find the other two roots.
If x, y ∈ R, then the determinant ∆ = `|(cosx, -sinx, 1),(sinx, cosx, 1),(cos(x + y), -sin(x + y), 0)|` lies in the interval.
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.
Prove tha `|("bc" - "a"^2, "ca" - "b"^2, "ab" - "c"^2),("ca" - "b"^2, "ab" - "c"^2, "bc" - "a"^2),("ab" - "c"^2, "bc" - "a"^2, "ca" - "b"^2)|` is divisible by a + b + c and find the quotient.
Let f(t) = `|(cos"t","t", 1),(2sin"t", "t", 2"t"),(sin"t", "t", "t")|`, then `lim_("t" - 0) ("f"("t"))/"t"^2` is equal to ______.
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 x, y, z are all different from zero and `|(1 + x, 1, 1),(1, 1 + y, 1),(1, 1, 1 + z)|` = 0, then value of x–1 + y–1 + z–1 is ______.
There are two values of a which makes determinant, ∆ = `|(1, -2, 5),(2, "a", -1),(0, 4, 2"a")|` = 86, then sum of these number is ______.
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 = ______.
`"A" = abs ((1/"a", "a"^2, "bc"),(1/"b", "b"^2, "ac"),(1/"c", "c"^2, "ab"))` is equal to ____________.
If A = `[(1,0,0),(2,"cos x","sin x"),(3,"sin x", "-cos x")],` then det. A is equal to ____________.
Find the minor of the element of the second row and third column in the following determinant `[(2,-3,5),(6,0,4),(1,5,-7)]`
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
The value of determinant `|(sin^2 13°, sin^2 77°, tan135°),(sin^2 77°, tan135°, sin^2 13°),(tan135°, sin^2 13°, sin^2 77°)|` is
Value of `|(2, 4),(-1, 2)|` is
