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
Find values of x, if `|[2,3],[4,5]|=|[x,3],[2x,5]|`
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.
If A is a 3 × 3 matrix, \[\left| A \right| \neq 0\text{ and }\left| 3A \right| = k\left| A \right|\] then write the value of k.
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k.
Which of the following is not correct?
If A is a matrix of order 3 and |A| = 8, then |adj A| = __________ .
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
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)|` = ______.
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 A is invertible matrix of order 3 × 3, then |A–1| ______.
`|(0, xyz, x - z),(y - x, 0, y z),(z - x, z - y, 0)|` = ______.
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, B, and C be the three square matrices such that A = B + C, then Det A is equal to
The value of the determinant `abs ((1,0,0),(2, "cos x", "sin x"),(3, "sin x", "cos x"))` is ____________.
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)]`
Let A be a square matrix of order 2 x 2, then `abs("KA")` is equal to ____________.
Find the 5th term of expansion of `(x^2 + 1/x)^10`?
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
Value of `|(2, 4),(-1, 2)|` is
