Advertisements
Advertisements
Question
If A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]` is a matrix such that AAT = 9I, find the values of x and y
Solution
A = `[(1, 2, 2),(2, 1, -2),(x, 2, y)]`
AT = `[(1, 2, x),(2, 1, 2),(2, -2, y)]`
Given AAT = 9I
`[(1, 2, 2),(2, 1, -2),(x, 2, y)] [(1, 2, x),(2, 1, 2),(2, -2, y)] = 9[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
`[(1 + 4 + 4, 2 + 2 - 4, x + 4 + 2y),(2 + 2 - 4, 4 + 1 + 4, 2x + 2 - 2y),(x + 4 + 24, 2x + 2 - 2y, x^2 + 4 + y^2)] = [(9, 0, 0),(0, 9, 0),(0, 0, 9)]`
Equating the corresponding entries
x + 4 + 2y = 0 .......(1)
2x + 2 – 2y = o .......(2)
x + 4 + 2y = 0 .......(3)
2x + 2 – 2y = 0 .......(4)
x2 + 4 + y2 = 9 .......(5)
(1) ⇒ x + 2y + 4 = 0
(2) ⇒ x – y + 1 = 0
– ing 3y + 3 = 0
y = `(-3)/3`
= – 1
Substituting the value of y in equation (1) we have
x + 4 + 2x – 1 = 0
x + 4 – 2 = 0
⇒ x = – 2
Substituting x = – 2 and y = – 1 in equation (5) we have
(5) ⇒ (– 2)2 + 4 + (– 1)2 = 9
4 + 4 + 1 = 9
9 = 9
∴ The required values of x and y are
x = – 2 and y = – 1
APPEARS IN
RELATED QUESTIONS
If a matrix has 18 elements, what are the possible orders it can have? What if it has 6 elements?
Construct a 3 × 3 matrix whose elements are given by aij = `("i" + "j")^3/3`
Find the values of x, y and z from the following equation
`[(12, 3),(x, 3/2)] = [(y, z),(3, 5)]`
Find x and y if `x[(4),(-3)] + y[(-2),(3)] = [(4),(6)]`
Solve for x, y : `[(x^2),(y^2)] + 2[(-2x),(-y)] = [(5),(8)]`
Let A = `[(1, 2),(1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, 2)]` Show that (A – B)C = AC – BC
Find the values of p, q, r, and s if
`[("p"^2 - 1, 0, - 31 - "q"^3),(7, "r" + 1, 9),(- 2, 8, "s" - 1)] = [(1, 0, -4),(7, 3/2, 9),(-2, 8, -pi)]`
Consider the matrix Aα = `[(cos alpha, - sin alpha),(sin alpha, cos alpha)]` Find all possible real values of α satisfying the condition `"A"_alpha + "A"_alpha^"T"` = I
Give your own examples of matrices satisfying the following conditions:
A and B such that AB = 0 and BA ≠ 0
If AT = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following
(BT)T = B
Choose the correct alternative:
If aij = (3i – 2j) and A = [aij]3 × 2 is
Choose the correct alternative:
If A and B are two matrices such that A + B and AB are both defined, then
Choose the correct alternative:
If A = `[(1, 2, 2),(2, 1, -2),("a", 2, "b")]` is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the ordered pair (a, b) is equal to
Choose the correct alternative:
If the points (x, – 2), (5, 2), (8, 8) are collinear, then x is equal to
Choose the correct alternative:
A root of the equation `|(3 - x, -6, 3),(-6, 3 - x, 3),(3, 3, -6 - x)|` = 0 is
Choose the correct alternative:
If a ≠ b, b, c satisfy `|("a", 2"b", 2"c"),(3, "b", "c"),(4, "a", "b")|` = 0, then abc =
Let P = `[(3, -1, -2),(2, 0, alpha),(3, -5, 0)]`, where α ∈ R. Suppose Q = [qij] is a matrix satisfying PQ = kI3 for some non-zero k ∈ R. If q23 = `-k/8` and |Q| = `k^2/2`, then α2 + k2 is equal to ______.
Let A = `[(cosα, -sinα),(sinα, cosα)]`, α ∈ R such that A32 = `[(0, -1),(1, 0)]`. Then a value of α is ______.
If A = `[(1, 2),(2, 3)]` and A2 – kA – I2 = 0, then value of k is ______.
The number of matrices A = `[(a, b),(c, d)]`, where a, b, c, d ∈ {–1, 0, 1, 2, 3, ............, 10} such that A = A–1, is ______.
