Linear Algebra, 2020a

Section IV. Matrix Operations 253
Remark. This illustrates that in practice we often want to compute linear combinations
of rows and columns in a context where we really aren’t interested in any
associated linear maps.
3.32 Express this nonsingular matrix as a product of elementary reduction matrices.
⎛ ⎞
1 2 0
T = ⎝2 −1 0⎠
3 1 2
3.33 Express
( 1
) 0
−3 3
as the product of two elementary reduction matrices.
̌ 3.34 Prove that the diagonal matrices form a subspace of M n×n . What is its
dimension?
3.35 Does the identity matrix represent the identity map if the bases are unequal?
3.36 Show that every multiple of the identity commutes with every square matrix.
Are there other matrices that commute with all square matrices?
3.37 Prove or disprove: nonsingular matrices commute.
̌ 3.38 Show that the product of a permutation matrix and its transpose is an identity
matrix.
3.39 Show that if the first and second rows of G are equal then so are the first and
second rows of GH. Generalize.
3.40 Describe the product of two diagonal matrices.
̌ 3.41 Show that if G has a row of zeros then GH (if defined) has a row of zeros. Does
that work for columns?
3.42 Show that the set of unit matrices forms a basis for M n×m .
3.43 Find the formula for the n-th power of this matrix.
( ) 1 1
1 0
̌ 3.44 The trace of a square matrix is the sum of the entries on its diagonal (its
significance appears in Chapter Five). Show that Tr(GH) =Tr(HG).
3.45 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?
3.46 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.
3.47 Give an example of two matrices of the same rank and size with squares of
differing rank.
3.48 Matrix multiplication is performed often on computers. Researchers trying to
understand its performance, and improve on it, count the number of operations
that it takes.
(a) Definition 2.3 gives p i,j = g i,1 h 1,j + g i,2 h 2,j + ···+ g i,r h r,j . How many real
number multiplications are in that expression? Using it, how many do we need
for the product of a m×r matrix and a r×n matrix?