Question 101340
Start with the given matrix

{{{A=(matrix(2,3,1,2,0,0,2,1))}}}



Find the transpose of A by flipping each entry over the diagonal. In other words, the entry in the first row and second column switches with the entry in the second row and first column. So in general {{{a[ij]=a[ji]}}} where i and j are the row and column numbers


{{{A^T=(matrix(3,2,1,0,2,2,0,1))}}}



Now multiply the two matrices


{{{A*A^T=(matrix(2,3,1,2,0,0,2,1))(matrix(3,2,1,0,2,2,0,1))=(matrix(2,2,5,4,4,5))}}}




Now find the determinant of the previous matrix. Remember, if {{{A=(matrix(2,2,a,b,c,d))}}}, the determinant of matrix A is {{{det(A)=a*d-b*c}}}


{{{det(A)=5*5-4*4=25-16=9}}}



Now find the inverse through this formula:


{{{A=(matrix(2,2,a,b,c,d))}}}===>{{{A^-1=(1/det(A))(matrix(2,2,d,-b,-c,a))}}}



{{{A^-1=(1/9)(matrix(2,2,5,-4,-4,5))=(matrix(2,2,5/9,-4/9,-4/9,5/9))}}}