# Question on Matrices?

Show that A^1 =

(2/3 -1/3 1)

(0 0 1)

(1/6 1/6 -1/2)

Update:

Sorry... should be A^ -1 = the matrix

Relevance
• 10 years ago

What is A in the first place? It looks like some kind of stochastic matrix, but since there is a negative entry and the sum in each row isn't one, I quit.

But wait, it looks like you are looking for the inverse of the matrix

A={{2/3,-1/3,1},{0,0,1},{1/6,1/6,-1/2}}

For a start, this matrix has determinant equal -1 (2/3(1/6)+1/6(1/3)) = -(1/9 + 1/18) = -3/18 = -1/6.

There are man ways to find its inverse, you can

1) augment it with the 3x3 identity matrix then do elementary row operations so that the augmented matrix is in reduced row echelon form (rref), then the right hand side gives you the inverse;

2) Find the cofactor matrix (the matrix whose entries are co-factors, (i.e determinant of a 2x2 minor matrix with +1 or -1) then divide the cofactor matrix by the determinant -1/6

3)put {{2/3,-1/3,1},{0,0,1},{1/6,1/6,-1/2}} into Wolram Alpha's input and click the equal sign and

the inverse is {1,0,2},{-1,3,4},{0,1,0}

Source(s): David Lay or Gilbert Strang