Problem 1 — The Mechanics of Matrix Multiplication
Recall that if has matrix and has matrix , then the composition has matrix , the matrix whose entry is the dot product of row of with column of .
Let
(a) Compute both and . Note that they are not even the same size. Then explain why the product is not defined.
(b) A product is defined only when the number of columns of equals the number of rows of . Explain this rule purely in terms of the composition it represents: which space must simultaneously be the codomain of one map and the domain of the other?
(c) Prove that matrix multiplication is associative: whenever the sizes are compatible, . (Do not expand the triple sum. Instead, let be the matrices of linear maps and use that the product of matrices is the matrix of a composition, together with the fact that composition of functions satisfies .)
Problem 2 — Order Matters: Rotations and Reflections
Let be rotation by counterclockwise and be reflection across the -axis:
(a) Compute and . Verify they are different, so .
(b) Each of and is itself the matrix of a reflection across some line through the origin. Identify the two lines by computing where each product sends and . Explain geometrically why “rotate then reflect” and “reflect then rotate” land on different lines.
(c) Find all matrices that commute with , i.e. satisfy . (Write , compute both products, and compare entries.) Show that the answer is exactly the matrices of the form — the “rotation–scaling” matrices, which you may recognize as the matrices acting like the complex number . Why is it geometrically believable that these, and only these, commute with a rotation?
Problem 3 — Composition in an Abstract Space: the Derivative Squared
Matrix multiplication computes composition in any vector spaces, not just . Let be differentiation, , where is the space of real polynomials of degree at most .
(a) Find the matrix of in the standard basis (apply to each basis vector and record the coordinate columns). Then compute the matrix product .
(b) Let be the second-derivative map, . Compute the matrix of directly from its action on and confirm it equals the product from (a). What are and , and why does the kernel grow under composition?
(c) Prove in general: for linear maps and , . Give an explicit example (you may use ) where the inclusion is strict.
Problem 4 — Powers of a Matrix: Projections and Shears
For a square matrix we write ( factors), and . A matrix is idempotent if , and nilpotent if for some .
Consider
(a) Compute ; ; and . Which of the three matrices is idempotent, and which is nilpotent?
(b) Conjecture a formula for and for (). Interpret each geometrically: what does repeatedly shearing, or repeatedly projecting, do?
(c) Prove your formula for by induction on . Then prove the structural fact behind idempotents: if , then every splits as with and .
Problem 5 — The Identity, and Which Matrices Commute
The identity matrix has ‘s on the diagonal and ‘s elsewhere; it is the matrix of the identity map , so for every matrix .
(a) Verify for .
(b) Let . Find all matrices satisfying . (Set up the equation and compare entries.) What kind of matrices appear?
(c) Prove: if a matrix commutes with every matrix, then for some scalar (a scalar matrix). (Hint: it is enough that commute with and .)
Problem 6 — The Trace
The trace of a square matrix is the sum of its diagonal entries: . Let
(a) Compute and (note they have different sizes), then compute and . Are the two traces equal?
(b) Show by a small example that the trace is not multiplicative: find matrices with . Is the trace additive, i.e. is always?
(c) Prove that whenever both products are defined (so is and is ). (Write each side as a double sum and swap the order of summation.)
Problem 7 — The Transpose
The transpose of an matrix is the matrix with : rows become columns. A square matrix is symmetric if . Let
(a) Compute , then compute separately, and verify they agree. Check that is not even defined. TYPO: Here it is actually defined.
(b) Show that for any matrix , the product is square and symmetric. For a general matrix, how many independent entries does a symmetric matrix have — i.e. what is the dimension of the space of symmetric matrices?
(c) Prove the reversal rule for any compatible , working entry-by-entry from the definitions of transpose and product.
Problem 8 — The Inverse of a Matrix
A square matrix is invertible if there is a matrix with ; then is the matrix of the inverse map, and it undoes . For a matrix the number (called the determinant of ) decides the matter: when it is nonzero,
(a) For , find in two ways: by solving for in terms of , and by the boxed formula. Verify .
(b) Show has no inverse by computing , and confirm this concretely by exhibiting two different vectors with the same image under . Why does a single such pair already rule out an inverse?
(c) Prove two facts. First, the inverse is unique: if and then . Second, if and are both invertible then so is , with (the order reverses — “socks and shoes”).
Problem 9 — Invertible Means Trivial Kernel
A square matrix is invertible if there is a matrix with . For a square matrix this is tightly bound to the kernel of the map .
(a) For each of
row-reduce and find . State which matrix has trivial kernel.
(b) Explain why, for a square matrix, the following are equivalent: ; the columns of are linearly independent; is surjective. Which theorem lets you pass from “injective” to “surjective” for a square matrix?
(c) Prove: for a square matrix , the map has if and only if is invertible. (You may use the rank-nullity theorem and the fact that a bijective linear map has a linear inverse, whose matrix is then .)
Problem 10 — Solving Systems with the Inverse
A square matrix is invertible if there is a matrix with . Let
(a) First check that the given really satisfies . Then solve for , , and by computing .
(b) When you must solve for many different right-hand sides , what is the advantage of computing once? If instead were not invertible, what would that say about the existence and uniqueness of solutions, in the language of surjectivity and injectivity?
(c) Prove: if is invertible then has the unique solution for every . (Show this works, and that no other vector can.)
Problem 11 — The Determinant as (Signed) Area
For a matrix, define the number . The map with this matrix sends the unit square — spanned by and — to the parallelogram spanned by the columns and . This problem uncovers what the number actually measures. (The area of the parallelogram spanned by two vectors and is .)
(a) For each of
compute the determinant, and separately compute the area of the image parallelogram (the one spanned by the two columns). Check in each case that the area equals . Which of the three maps preserves area?
(b) Two discoveries.
Orientation. Compute and (the second swaps the columns of the first). The two image parallelograms have the same area, yet the determinants differ in sign — so is a signed area, the sign recording whether the map preserves or reverses orientation. What is for the reflection across the -axis, and is its sign consistent with “a reflection flips orientation”?
Areas multiply. Since is the factor by which area scales, applying then should scale area by . Test this with and : compute , , and , and confirm .
(c) Prove directly from the definition of the kernel that the map with matrix is injective if and only if . Interpret this geometrically using part (a): the image parallelogram has nonzero area exactly when the map does not collapse the plane onto a line.
Problem 12 — Eigenvalues: Directions a Map Only Stretches
A nonzero vector is an eigenvector of a square matrix with eigenvalue if — the map stretches by the factor without changing its direction. Equivalently, is a nonzero vector in , so is an eigenvalue exactly when has a nonzero solution.
(a) Let . For to be an eigenvalue, the two rows of must be linearly dependent (otherwise the kernel is trivial). Cross-multiplying the proportionality of the rows gives . Solve this for the two eigenvalues, then find an eigenvector for each by computing .
(b) Check that your two eigenvectors form a basis of . Using only , compute and . Then explain how to compute for an arbitrary once it is written in the basis . (This is the payoff of a basis of eigenvectors: high powers become easy.)
(c) Prove two facts. First, if then for every integer . Second, eigenvectors belonging to distinct eigenvalues are linearly independent — prove this for two eigenvectors with .
Problem 13 — Capstone: When Is a Square Matrix Invertible?
A square matrix is invertible if there is a matrix with . This problem assembles several conditions on the map :
(i) is invertible; (ii) ; (iii) the columns of form a basis of ; (iv) has a unique solution for every ; (v) is surjective ().
Apply them to
(a) Verify conditions (ii) and (iii) explicitly for (row-reduce and inspect the kernel and the columns), and show that each of (i)–(v) fails for .
(b) Explain the chain of reasoning tying these conditions together into one equivalence. Which result lets you move from "" (injective) to "" (surjective) for a square matrix?
(c) Prove two of the equivalences from scratch: (ii) (iii), and (i) (ii). State clearly which prior result you invoke at each step.