If A = [[1,-1,0,0],[2,-3,0,0],[0,0,1,-1],[0,0,-2,3]] then A^-1 = ?

Expert Answers
rakesh05 eNotes educator| Certified Educator

Given matrix is `A=[[1,-1,0,0],[2,-3,0,0],[0,0,1,-1],[0,0,-2,3]]` .

Now to find the inverse of the matrix A we write the matrix equation as `AI=A.`  where I is the `4xx4`  identity matrix. Now by applying elementary row operations reduce the matrix A on the right hand side to identity matrix I.

i.e.   `AB=I` .  Here the matrix B will be the inverse of the matrix A.

So,    `A[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]=[[1,-1,0,0],[2,-3,0,0],[0,0,1,-1],[0,0,-2,3]]`

by `R_2->2R_1-R_2` we get


by   `R_1->R_1+R_2`  and  `R_4->2R_3+R_4`  we get


by  `R_3->R_3+R_4`  we get

     `A[[3,-1,0,0],[2,-1,0,0],[0,0,3,1],[0,0,2,1]]=[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]` .

Now we see that the matrix on right hand side is the identity. So the matrix `B=[[3,-1,0,0],[2,-1,0,0],[0,0,3,1],[0,0,2,1]]`  is the required inverse of the matrix A.

         So `A^-1=[[3,-1,0,0],[2,-1,0,0],[0,0,3,1],[0,0,2,1]]` .


pramodpandey | Student

A = [[1,-1,0,0],[2,-3,0,0],[0,0,1,-1],[0,0,-2,3]]





`XE=I, XG=0,YF=0,YH=I`