Iwas asked to solve an invertible matrix, found the inverse but having trouble putting it into a product of elementary matrices. can anyone ?
a^-1 = [-9/2 7/2]
[ 4 -3]

I'm guessing you were originally told to find the inverse of

and you've found the inverse to be

I'm also guessing that "product of elementary matrices" includes the decomposition of into lower and upper triangular as well as diagonal matrices.

First thing I would do would be eliminate the fractions by multiplying the first row of by 2. In matrix form, this is done by multiplying by

which you can interpret as "multiply the first row by 2 and leave the second row alone":

Next, we make the matrix on the right side upper-triangular by eliminating the entry in row 2, column 1. This is done via the product

which you can interpret as "leave the first row alone, and replace row 2 by 4(row 1) + 9(row 2)".

Lastly, multiply both sides by the inverses of all matrices as needed to isolate on the left side. That is,

For two invertible matrices and , we have , so that

Compute the remaining inverses:

So we have