Question:
If A = yz x zx y xy z Where x, y and z are integers, is an orthogonal matrix, then what is the value of x2 + y2 + z2?
If A = yz x zx y xy z Where x, y and z are integers, is an orthogonal matrix, then what is the value of x2 + y2 + z2?
Options:
- a) 0
- b) 1
- c) 4
- d) 14
- a) 0
- b) 1
- c) 4
- d) 14
correct answer : b)
Explanation:
For orthogonal matrix.
AT = A – 1 Þ AAT = I
A2 = I (\ A is symmetric)
yz x
zx y
xy z
yz x
zx y
xy z
=
10 0
01 0
00 1
yz xy zz xx yx yy zx z
yz zx xy xy zx yy zz x
xy yz zx xy
22 2
22 2
++ ++ ++
++ ++ ++
++ +++ ++
yz zx xy z
22 2
=
10 0
01 0
00 1
x2 + y2 + z2 = 1
For orthogonal matrix.
AT = A – 1 Þ AAT = I
A2 = I (\ A is symmetric)
yz x
zx y
xy z
yz x
zx y
xy z
=
10 0
01 0
00 1
yz xy zz xx yx yy zx z
yz zx xy xy zx yy zz x
xy yz zx xy
22 2
22 2
++ ++ ++
++ ++ ++
++ +++ ++
yz zx xy z
22 2
=
10 0
01 0
00 1
x2 + y2 + z2 = 1