Ex 3.2, 18 - Chapter 3 Class 12 Matrices

Transcript

Ex 3.2, 18 If A =[ 8(0& tan /2 " " @tan /2 " " &0)] and I is the identity matrix of order 2, Show that I + A = ( I A)[ 8(cos & sin @sin &cos )] Given I the identity matrix of order 2 i.e. I = [ 8(1&0@0&1)] Taking L.H.S. I + A = [ 8(1&0@0&1)] + [ 8(0& tan /2 " " @tan /2 " " &0)] = [ 8(1+0&0 tan /2 " " @0+tan /2 " " &1+0)] = [ 8(1& tan /2 " " @tan /2 " " &1)] Taking R.H.S (I A) [ 8(cos & sin @sin &cos )] = (" " [ 8(1&0@0&1)] [ 8(0& tan /2 " " @tan /2 " " &0)]) [ 8(cos & sin @sin &cos )] = [ 8(1 0&0+tan /2 " " @0 tan /2 " " &1)][ 8(cos & sin @sin &cos )] = [ 8(1&tan /2 " " @ tan /2 " " &1)][ 8(cos & sin @sin &cos )] = [ 8(1&tan /2 " " @ tan /2 " " &1)] [ 8((1 2 /2)/(1 + 2 /2) " " &( 2 tan /2 )/(1 + 2 /2) " " @(2 tan /2 )/(1 + 2 /2) " " &(1 2 /2)/(1 + 2 /2) " " )] = [ 8(1((1 2 /2)/(1 + 2 /2))"+ " /2 ((2 /2)/(1 + 2 /2))" " &1(( 2 /2)/(1+ 2 /2))"+(" /2) ((1 2 /2)/(1 + 2 /2))" " @ /2 ((1 2 /2)/(1 + 2 /2))" +1" ((2 /2)/(1 + 2 /2))& /2 (( 2 2 /2)/(1 + 2 /2))" +1" ((1 2 /2)/(1 + 2 /2)) )] = [ 8((1 2 /2)/(1 + 2 /2) " +" (2 2 /2)/(1 + 2 /2) " " &( 2 tan /2 " +" tan /2 3 /2)/(1 + 2 /2) " " @( tan /2 (1 2 /2))/(1 + 2 /2) " +" (2 tan /2 )/(1 + 2 /2)&(2 2 /2)/(1+ 2 /2)+ " " (1 2 /2)/(1 + 2 /2))] = [ 8((1 2 /2 + 2 2 /2)/(1 + 2 /2) " " &( 2 tan /2 " + " tan /2 3 /2)/(1 + 2 /2) " " @( tan /2 + 3 /2+2 tan /2 )/(1 + 2 /2) " " &" " (2 2 /2 + 1 2 /2)/(1 + 2 /2))] = [ 8((1 + 2 /2 )/(1 + 2 /2) " " &( tan /2 " " (1 + 2 /2) )/(1 + 2 /2) " " @( tan /2 (1 + 2 /2))/(1 + 2 /2) " " &" " (1 + 2 /2 )/(1 + 2 ( )/2))] = [ 8(1& tan /2 " " @tan /2 " " &1)] = R.H.S.