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प्रश्न
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
उत्तर
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
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