Page 72 - A Course in Linear Algebra with Applications

56 Chapter Two: Systems of Linear Equations

2. Express the second matrix in Exercise 1 as a product of
elementary matrices and its reduced column echelon form.
3. Find the normal form of each matrix in Exercise 1.

4. Find the inverses of the three types of elementary matrix,
and observe that each is elementary and corresponds to the
inverse row operation.
5. What is the maximum number of column operations needed
in general to put an n x n matrix in column echelon form and
in reduced column echelon form?
6. Compute the inverses of the following matrices if they exist:

2 - 3 1 \ / 2 1 7
1 0 - 1 4 10
2 ; (c)
0 - 1 - 3 / \ 3 2 12

7. For which values of t does the matrix

6 - 1 1
t 0 1
0 1 t

not have an inverse?
8. Give necessary and sufficient conditions for an upper tri-
angular matrix to be invertible.
9. Show by an example that if an elementary column opera-
tion is applied to the augmented matrix of a linear system, the
resulting linear system need not be equivalent to the original
one.
10. Prove that the number of elementary row operations
2
needed to find the inverse of an n x n matrix is at most n .