Linear Algebra, 2020a

Section IV. Matrix Operations 243
2.22 [Cleary] Match each type of matrix with all these descriptions that could fit,
say ‘None’ if it applies: (i) can be multiplied by its transpose to make a 1×1 matrix,
(ii) can represent a linear map from R 3 to R 2 that is not onto, (iii) can represent
an isomorphism from R 3 to P 2 .
(a) a 2×3 matrix whose rank is 1
(b) a 3×3 matrix that is nonsingular
(c) a 2×2 matrix that is singular
(d) an n×1 column vector
2.23 Show that composition of linear transformations on R 1 is commutative. Is this
true for any one-dimensional space?
2.24 Why is matrix multiplication not defined as entry-wise multiplication? That
would be easier, and commutative too.
2.25 (a) Prove that H p H q = H p+q and (H p ) q = H pq for positive integers p, q.
(b) Prove that (rH) p = r p · H p for any positive integer p and scalar r ∈ R.
̌ 2.26 (a) How does matrix multiplication interact with scalar multiplication: is
r(GH) =(rG)H? IsG(rH) =r(GH)?
(b) How does matrix multiplication interact with linear combinations: is F(rG +
sH) =r(FG)+s(FH)? Is(rF + sG)H = rFH + sGH?
2.27 We can ask how the matrix product operation interacts with the transpose
operation.
(a) Show that (GH) T = H T G T .
(b) A square matrix is symmetric if each i, j entry equals the j, i entry, that is, if
the matrix equals its own transpose. Show that the matrices HH T and H T H are
symmetric.
̌ 2.28 Rotation of vectors in R 3 about an axis is a linear map. Show that linear maps
do not commute by showing geometrically that rotations do not commute.
2.29 In the proof of Theorem 2.12 we used some maps. What are the domains and
codomains?
2.30 How does matrix rank interact with matrix multiplication?
(a) Can the product of rank n matrices have rank less than n? Greater?
(b) Show that the rank of the product of two matrices is less than or equal to the
minimum of the rank of each factor.
2.31 Is ‘commutes with’ an equivalence relation among n×n matrices?
2.32 (We will use this exercise in the Matrix Inverses exercises.) Here is another
property of matrix multiplication that might be puzzling at first sight.
(a) Prove that the composition of the projections π x ,π y : R 3 → R 3 onto the x and
y axes is the zero map despite that neither one is itself the zero map.
(b) Prove that the composition of the derivatives d 2 /dx 2 ,d 3 /dx 3 : P 4 → P 4 is the
zero map despite that neither is the zero map.
(c) Give a matrix equation representing the first fact.
(d) Give a matrix equation representing the second.
When two things multiply to give zero despite that neither is zero we say that each
is a zero divisor.
2.33 Show that, for square matrices, (S + T)(S − T) need not equal S 2 − T 2 .