Advertisements
Advertisements
प्रश्न
If A = `[(1/2, alpha),(0, 1/2)]`, prove that `sum_("k" = 1)^"n" det("A"^"k") = 1/3(1 - 1/4)`
Advertisements
उत्तर
A = `[(1/2, alpha),(0, 1/2)]`
|A| = `|(1/2,alpha),(0, 1/2)|`
= `1/4 - 0`
= `1/4`
A2 = A × A
= `[(1/2, alpha),(0, 1/2)] [(1/2, alpha),(0, 1/2)]`
= `[(1/4, alpha),(0, 1/4)]`
|A2| = `|(1/4, alpha),(0, 1/4)|`
=`1/4 xx 1/4 - 0`
= `(1/4)^2`
= `1/4^2`
|Ak| = `1/4^"k"`
So `sum_("k" = 1)^"n" det("A"^"k") = 1/4 + 1/4^2 + 1/4^3 + ...... + 1/4^"n"`
Which is a G.P with a `1/4` and r = `1/4`
∴ Sn = `("a"(1 - "r"^"n"))/(1 - "r")`
= `(1/4[1 - (1/4)^"n"])/(1 - 1/4)`
= `(1/4[1 - 1/4^"n"])/(3/4)`
= `1/4 xx 4/3[1 - 1/4^"n"]`
= `1/3[1 - 1/4^"n"]`.
APPEARS IN
संबंधित प्रश्न
Without expanding the determinant, prove that `|("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)|` = 0
Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 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|
Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0
Verify that det(AB) = (det A)(det B) for A = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]` and B = `[(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`
Using cofactors of elements of second row, evaluate |A|, where A = `[(5, 3, 8),(2, 0, 1),(1, 2, 3)]`
Identify the singular and non-singular matrices:
`[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`
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 ⌊.⌋ 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)]`
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
Find the area of the triangle with vertices at the point given is (1, 0), (6, 0), (4, 3).
Choose the correct option:
Let `|(0, sin theta, 1),(-sintheta, 1, sin theta),(1, -sin theta, 1 - a)|` where 0 ≤ θ ≤ 2n, then
Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ2 is equal to ______.
`|("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 ______.
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 ______.
