Find the inverse of the matrix with elements a11=2,a12=3,a13=1,a21=3,a22=7,a23=2,a31=2,a2=4,a33=1 using elementary row and elementary column tranformation method.Sir,Can you suggest any way to do the above method more easily?
(for 3 by 3 matrix)
 
 

Asked by Rajeev | 23rd Jun, 2015, 03:02: PM

Expert Answer:

C o n s i d e r space t h e space m a t r i x space A equals open square brackets table row 2 3 1 row 3 7 2 row 2 4 1 end table close square brackets W e space n e e d space t o space f i n d space t h e space i n v e r s e space o f space t h e space m a t r i x space A. L e t space u s space a u g m e n t space t h e space i d e n t i t y space m a t r i x space w i t h space t h e space g i v e n space m a t r i x. open square brackets table row 2 3 1 row 3 7 2 row 2 4 1 end table close square brackets open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets  A p p l y space R subscript 1 rightwards arrow fraction numerator space R subscript 1 over denominator 2 end fraction open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 3 7 2 row 2 4 1 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row 0 1 0 row 0 0 1 end table close square brackets  A p p l y space R subscript 2 rightwards arrow R subscript 2 minus 3 R subscript 1 open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 cell 5 over 2 end cell cell 1 half end cell row 2 4 1 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell fraction numerator negative 3 over denominator 2 end fraction end cell 1 0 row 0 0 1 end table close square brackets  A p p l y space R subscript 3 rightwards arrow R subscript 3 minus 2 R subscript 1 open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 cell 5 over 2 end cell cell 1 half end cell row 0 1 0 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell fraction numerator negative 3 over denominator 2 end fraction end cell 1 0 row cell negative 1 end cell 0 1 end table close square brackets  A p p l y space R subscript 2 rightwards arrow R subscript 2 cross times 2 over 5 open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 1 cell 1 fifth end cell row 0 1 0 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell fraction numerator negative 3 over denominator 5 end fraction end cell cell 2 over 5 end cell 0 row cell negative 1 end cell 0 1 end table close square brackets  A p p l y space R subscript 3 rightwards arrow R subscript 3 minus R subscript 2 open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 1 cell 1 fifth end cell row 0 0 cell fraction numerator negative 1 over denominator 5 end fraction end cell end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell fraction numerator negative 3 over denominator 5 end fraction end cell cell 2 over 5 end cell 0 row cell negative 2 over 5 end cell cell fraction numerator negative 2 over denominator 5 end fraction end cell 1 end table close square brackets  A p p l y space R subscript 3 rightwards arrow R subscript 3 cross times open parentheses negative 5 close parentheses open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 1 cell 1 fifth end cell row 0 0 1 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell fraction numerator negative 3 over denominator 5 end fraction end cell cell 2 over 5 end cell 0 row 2 2 cell negative 5 end cell end table close square brackets  A p p l y space R subscript 2 rightwards arrow R subscript 2 minus R subscript 3 over 5 open square brackets table row 1 cell 3 over 2 end cell cell 1 half end cell row 0 1 0 row 0 0 1 end table close square brackets open square brackets table row cell 1 half end cell 0 0 row cell negative 1 end cell 0 1 row 2 2 cell negative 5 end cell end table close square brackets  A p p l y space R subscript 1 rightwards arrow R subscript 1 minus R subscript 3 over 2 open square brackets table row 1 cell 3 over 2 end cell 0 row 0 1 0 row 0 0 1 end table close square brackets open square brackets table row cell fraction numerator negative 1 over denominator 2 end fraction end cell cell negative 1 end cell cell 5 over 2 end cell row cell negative 1 end cell 0 1 row 2 2 cell negative 5 end cell end table close square brackets  A p p l y space R subscript 1 rightwards arrow R subscript 1 minus fraction numerator 3 R subscript 2 over denominator 2 end fraction open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets open square brackets table row 1 cell negative 1 end cell 1 row cell negative 1 end cell 0 1 row 2 2 cell negative 5 end cell end table close square brackets  T h u s space t h e space i n v e r s e space o f space t h e space g i v e n space m a t r i x space i s space open square brackets table row 1 cell negative 1 end cell 1 row cell negative 1 end cell 0 1 row 2 2 cell negative 5 end cell end table close square brackets.

Answered by Vimala Ramamurthy | 24th Jun, 2015, 10:30: AM