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Asked by SanskarAgarwal86 5th March 2020, 1:09 PM
Answered by Expert
Answer:
To prove:
Determinant of ( x2+1, xy, xz // xy, y2+1, yz // xz, yz, z2+1 ) = 1+x2+y2+z2
open vertical bar table row cell x squared plus 1 end cell cell x y end cell cell x z end cell row cell x y end cell cell y squared plus 1 end cell cell y z end cell row cell x z end cell cell y z end cell cell z squared plus 1 end cell end table close vertical bar
M u l t i p l y i n g space R subscript 1 comma space R subscript 2 space a n d space R subscript 3 space b y space x comma space y space a n d space z space r e s p e c t i v e l y
equals fraction numerator 1 over denominator x y z end fraction open vertical bar table row cell x open parentheses x squared plus 1 close parentheses end cell cell x squared y end cell cell x squared z end cell row cell x y squared end cell cell y open parentheses y squared plus 1 close parentheses end cell cell y squared z end cell row cell x z squared end cell cell y z squared end cell cell z open parentheses z squared plus 1 close parentheses end cell end table close vertical bar
equals fraction numerator x y z over denominator x y z end fraction open vertical bar table row cell x squared plus 1 end cell cell x squared end cell cell x squared end cell row cell y squared end cell cell y squared plus 1 end cell cell y squared end cell row cell z squared end cell cell z squared end cell cell z squared plus 1 end cell end table close vertical bar
R subscript 1 rightwards arrow R subscript 1 plus R subscript 2 plus R subscript 3
equals open vertical bar table row cell x squared plus y squared plus z squared plus 1 end cell cell x squared plus y squared plus z squared plus 1 end cell cell x squared plus y squared plus z squared plus 1 end cell row cell y squared end cell cell y squared plus 1 end cell cell y squared end cell row cell z squared end cell cell z squared end cell cell z squared plus 1 end cell end table close vertical bar
C subscript 2 rightwards arrow C subscript 2 minus C subscript 1 space space a n d space space space C subscript 3 rightwards arrow C subscript 3 minus C subscript 1
equals open vertical bar table row cell x squared plus y squared plus z squared plus 1 end cell 0 0 row cell y squared end cell 1 0 row cell z squared end cell 0 1 end table close vertical bar
E x p a n d i n g space a l o n g space R subscript 1
equals x squared plus y squared plus z squared plus 1
Answered by Expert 6th March 2020, 3:07 PM
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