Example 24 - Chapter 3 Class 12 Matrices

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Example 24 Obtain the inverse of the following matrix using elementary operations A = [ 8(0&1&2@1&2&3@3&1&1)] Given A = [ 8(0&1&2@1&2&3@3&1&1)] We know that A = IA [ 8(0&1&2@1&2&3@3&1&1)] = [ 8(1&0&0@0&1&0@0&0&1)] A R1 R2 [ 8( &2&3@0&1&2@3&1&1)] = [ 8(0&1&0@1&0&0@0&0&1)] A R3 R3 3R1 [ 8(1&2&3@0&1&2@ ( )&1 3(2)&1 3(3))] = [ 8(0&1&0@1&0&0@0 3(0)&0 3(1)&1 3(0))]A [ 8(1&2&3@0&1&2@ & 5& 8)] = [ 8(0&1&0@1&0&0@0& 3&1)] R1 R1 2R2 [ 8(1 2(0)& ( )&3 2(2)@0&1&2@0& 5& 8)] = [ 8(0 2(1)&1 2(0)&0 2(0)@1&0&0@0& 3&1)]A [ 8(1& & 1@0&1&2@0& 5& 8)] = [ 8( 2&1&0@1&0&0@0& 3&1)] A R3 R3 + 5R2 [ 8(1&0& 1@0&1&2@0+5(0)& + ( )& 8+5(2))] = [ 8( 2&1&0@1&0&0@0+5(1)& 3+5(0)&1+5(0))] A [ 8(1&0& 1@0&1&2@0& &2)] = [ 8( 2&1&0@1&0&0@5& 3&1)] A R3 1/2 R3 [ 8(1&0& 1@0&1&2@0/2&0/2& / )] = [ 8( 2&1&0@1&0&0@5/2&( 3)/2&1/2)] A R1 R1 + R3 [ 8(1+0&0+0& + @0&1&2@0&0&1)]=[ 8( 2+5/2&1+(( 3)/2)&0+1/2@1&0&0@5/2&( 3)/2&1/2)] A [ 8(1&0& @0&1&2@0&0&1)] = [ 8(1/2&( 1)/2&1/2@1&0&0@5/2&( 3)/2&1/2)] A R2 R2 2R3 [ 8(1&0&0@0 2(0)&1 2(0)& ( )@0&0&1)] = [ 8(1/2&( 1)/2&1/2@1 2(5/2)&0 2(( 3)/2)&0 2(1/2)@5/2&( 3)/2&1/2)]A [ 8(1&0&0@0&1& @0&0&1)] = [ 8(1/2&( 1)/2&1/2@ 4&3& 1@5/2&( 3)/2&1/2)] A I= [ 8(1/2&( 1)/2&1/2@ 4&3& 1@5/2&( 3)/2&1/2)] A This is similar to I = A-1 A Hence A-1 = [ 8(1/2&( 1)/2&1/2@ 4&3& 1@5/2&( 3)/2&1/2)]