Linear Algebra

Section IV. Matrix Operations 229
� 3.25 The need to take linear combinations of rows and columns in tables of numbers
arises often in practice. For instance, this table gives the number of hours of each
type done by each worker, and the associated pay rates. Use matrices to compute
the wages due.
regular overtime
Alan 40 12
Betty 35 6
Catherine 40 18
Donald 28 0
3.26 Find the product of this matrix with its transpose.
� �
cos θ − sin θ
sin θ cos θ
wage
regular $25.00
overtime $45.00
� 3.27 Prove that the diagonal matrices form a subspace of Mn×n. What is its
dimension?
3.28 Does the identity matrix represent the identity map if the bases are unequal?
3.29 Show that every multiple of the identity commutes with every square matrix.
Are there other matrices that commute with all square matrices?
3.30 Prove or disprove: nonsingular matrices commute.
� 3.31 Show that the product of a permutation matrix and its transpose is an identity
matrix.
3.32 Show that if the first and second rows of G are equal then so are the first and
second rows of GH. Generalize.
3.33 Describe the product of two diagonal matrices.
3.34 Write � �
1 0
−3 3
as the product of two elementary reduction matrices.
� 3.35 Show that if G hasarowofzerosthenGH (if defined) has a row of zeros.
Does that work for columns?
3.36 Show that the set of unit matrices forms a basis for Mn×m.
3.37 Find the formula for the n-th power of this matrix.
� �
1 1
1 0
� 3.38 The trace of a square matrix is the sum of the entries on its diagonal (its
significance appears in Chapter Five). Show that trace(GH) =trace(HG).
� 3.39 A square matrix is upper triangular if its only nonzero entries lie above, or
on, the diagonal. Show that the product of two upper triangular matrices is upper
triangular. Does this hold for lower triangular also?
3.40 A square matrix is a Markov matrix if each entry is between zero and one
and the sum along each row is one. Prove that a product of Markov matrices is
Markov.
� 3.41 Give an example of two matrices of the same rank with squares of differing
rank.
3.42 Combine the two generalizations of the identity matrix, the one allowing entires
to be other than ones, and the one allowing the single one in each row and
column to be off the diagonal. What is the action of this type of matrix?