Suppose that a, b, c are real numbers such that a + b + c = 1. If the matrix be an orthogonal matrix, then
atleast one of a, b, c is negative
|A| is negative
a3 + b3 + c3 – 3abc = 1
All of these
As we have learned
Circulant determinant -
The elements of the rows (or columns) are in cyclic arrangement
- wherein
eg:-
AAT = ATA = I. Also AT = A, so A2 = I A is involuntary matrix.
|A2| = |A|2 = 1 or |A| = ±1
But |A| =
|A| = ab + bc + ca – a2 – b2 – c2
a2 + b2 + c2 – ab – bc – ca 0
So |A| = -1. Hence a3 + b3 + c3 – 3abc = 1.
Again a2 + b2 + c2 – ab – bc – ca = 1 1 – 3(ab + bc + c(A) = 1, so ab + bc + ca = 0
atleast one of a, b and c is negative.
Option 1)
atleast one of a, b, c is negative
Option 2)
|A| is negative
Option 3)
a3 + b3 + c3 – 3abc = 1
Option 4)
All of these
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