178 CHAPTER 8. RANK OF A MATRIX
(a)
(1 2 00 1 7
)
(b)
1 0 0 00 0 1 20 0 0 0
(c)
1 1 0 0 0 50 0 1 2 0 40 0 0 0 1 3
5. Row reduce the following matrices to obtain the row reduced echelon form. List thepivot columns in the original matrix.
(a)
1 2 0 32 1 2 21 1 0 3
(b)
1 2 32 1 −23 0 03 2 1
(c)
1 2 1 3−3 2 1 03 2 1 1
6. Find the rank of the following matrices. If the rank is r, identify r columns in theoriginal matrix which have the property that every other column may be written asa linear combination of these. Also find a basis for the row and column spaces of thematrices.
(a)
1 2 03 2 12 1 00 2 1
(b)
1 0 04 1 12 1 00 2 0
(c)
0 1 0 2 1 2 20 3 2 12 1 6 80 1 1 5 0 2 30 2 1 7 0 3 4
(d)
0 1 0 2 0 1 00 3 2 6 0 5 40 1 1 2 0 2 20 2 1 4 0 3 2
(e)
0 1 0 2 1 1 20 3 2 6 1 5 10 1 1 2 0 2 10 2 1 4 0 3 1
7. Suppose A is an m× n matrix. Explain why the rank of A is always no larger thanmin(m,n) .
8. A matrix A is called a projection if A2 = A. Here is a matrix. 2 0 21 1 2−1 0 −1