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

