7. Derive and verify a Cayley-Hamilton formula for the inverse of an invertible 5 × 5 matrix

Consider the n ×n matrices A and B and their product AB(a) Show that if AB is invertible, then so is A.(Hint: There is a matrix W such that ABW = I. Why?) (b) Show that if AB is invertible, then so is B.

a). Explain what changes in your formula when L is a general invertible lower- triangular matrix and how it reveals that the inverse of a lower-triangular matrix is lower-triangular  b). give a concise proof that the inverse of an upper triangular matrix is upper triangular

prove that each type of elementary row operation on an m × n matrix is invertible by giving the inverse operation.

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists)