CEGEP Linear Algebra — Matrices and Matrix Algebra Worksheet

All 7 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Matrix Notation and Special Matrices

   Let $$M=\left[\begin{array}{cccc}3& -1& 0& 5\\ 2& 4& -6& 1\\ 0& 7& 2& -3\end{array}\right].$$

   1. State the size of $M$, and give the entries $m_{23}$ and $m_{32}$.
   2. Write out in full the $3\times 3$ matrix $B$ whose entries are given by $b_{ij}=2i-j$.
   3. Write down the $2\times 4$ zero matrix and the identity matrix $I_4$, and give the entries $(I_4{)}_{33}$ and $(I_4{)}_{23}$.
2. Q2Addition and Scalar Multiplication

   Let $$A=\left[\begin{array}{ccc}2& -1& 3\\ 0& 4& -2\end{array}\right],B=\left[\begin{array}{ccc}-3& 5& 1\\ 6& 0& -4\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 2& 2\end{array}\right].$$

   1. Compute $3A-2B$.
   2. Find the matrix $X$ satisfying $A+2X=B$.
   3. Explain in one sentence why $A+C$ is not defined.
3. Q3Matrix Multiplication

   Let $$A=\left[\begin{array}{ccc}2& -1& 0\\ 3& 4& 1\end{array}\right],B=\left[\begin{array}{cc}1& 5\\ -2& 0\\ 4& 3\end{array}\right].$$

   1. State the sizes of $AB$ and of $BA$.
   2. Compute $AB$.
   3. Compute the single entry $(BA{)}_{31}$ without computing the rest of $BA$.
4. Q4Properties and Failures of Matrix Algebra

   Matrix algebra keeps most of the rules of ordinary algebra and loses a few. This question is about the ones it loses. Throughout, $0$ denotes the $2\times 2$ zero matrix.

   1. With $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 0\\ 3& 1\end{array}\right]$, compute $AB$ and $BA$, and say which rule of ordinary algebra this refutes.
   2. With $C=\left[\begin{array}{cc}2& -6\\ -1& 3\end{array}\right]$ and $D=\left[\begin{array}{cc}3& 6\\ 1& 2\end{array}\right]$, compute $CD$. Which property of the real numbers has just failed?
   3. Let $E=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ and $F=\left[\begin{array}{cc}4& 7\\ 2& 3\end{array}\right]$. Show that $CE=CF$ even though $E\ne F$ and $C\ne 0$, and name the step of ordinary algebra that is therefore invalid for matrices.
   4. Prove or disprove: for all $2\times 2$ matrices $A$ and $B$, $(A+B{)}^2=A^2+2AB+B^2$. If it is false, give the correct expansion and say exactly which step of the usual argument breaks.
5. Q5The Transpose and Symmetric Matrices

   Let $$A=\left[\begin{array}{ccc}1& 4& -2\\ 0& 3& 5\end{array}\right],K=\left[\begin{array}{ccc}0& 5& -3\\ -5& 0& 2\\ 3& -2& 0\end{array}\right].$$

   1. Write $A^T$ and state its size.
   2. Compute $A{A}^T$ and confirm that it is symmetric.
   3. Show that $K$ is antisymmetric, and explain why every antisymmetric matrix must carry zeros all along its main diagonal.
   4. Show that the zero matrix is the only square matrix that is both symmetric and antisymmetric.
6. Q6Diagonal and Triangular Matrices

   Let $$D=\left[\begin{array}{ccc}3& 0& 0\\ 0& -2& 0\\ 0& 0& 1\end{array}\right],U=\left[\begin{array}{ccc}2& -1& 4\\ 0& 5& 3\\ 0& 0& -1\end{array}\right],L=\left[\begin{array}{ccc}1& 0& 0\\ 6& 2& 0\\ -3& 4& 5\end{array}\right].$$

   1. Classify each of $D$, $U$, $L$ as diagonal, upper triangular or lower triangular. One of them deserves more than one label; say which, and why.
   2. Compute $DU$ and $UD$. Describe in words what multiplying by a diagonal matrix on the left does, and what it does on the right.
   3. Compute $U^2$, then explain why the product of any two upper triangular $3\times 3$ matrices must again be upper triangular.
7. Q7Synthesis — drawing on several topics in this set

   Every square matrix splits into a symmetric part and an antisymmetric part. For a square matrix $A$, define $$S=\frac{1}{2}(A+A^T),K=\frac{1}{2}(A-A^T).$$

   1. For $A=\left[\begin{array}{cc}4& 2\\ 6& -2\end{array}\right]$, compute $S$ and $K$.
   2. Verify for this $A$ that $S$ is symmetric, that $K$ is antisymmetric, and that $S+K=A$.
   3. Prove that $S$ is symmetric and $K$ is antisymmetric for every square matrix $A$. You may use $(X+Y{)}^T=X^T+Y^T$, $(cX{)}^T=c{X}^T$ and ${\left(X^T\right)}^T=X$.

The 4 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Two indices, and their order is not decorative

Q1 starts with the notation, because every later argument in linear algebra is written in it. Part (a) asks for the size of a matrix and then for two entries whose indices are the reverse of each other — deliberately, since they sit in different places and hold different numbers.

Part (b) then defines a matrix by a formula instead of writing it out, which is the step that catches people the first time. Nothing is hidden in it: run i down the rows and j across the columns, evaluate the formula at each pair, and put the number in that position. Compute one entry away from the diagonal as a check — if the formula is not symmetric in i and j, positions (i, j) and (j, i) must disagree, and finding that they do is confirmation you have the convention the right way round.

Part (c) asks for the zero matrix of a stated size and the identity of a stated order, and then for one entry of the identity on the diagonal and one off it. That is the definition of the identity restated as arithmetic: 1 where i = j, 0 everywhere else. Notice that the zero matrix comes in every size, while the identity is square by definition and is labelled by its order alone.

Addition is entry by entry, and it is fussy about size

Q2 does the two operations that behave exactly as you would hope. A scalar multiplies every entry; a sum pairs entry (i, j) with entry (i, j). All the usual algebra survives here, which is why part (b) can be solved the way a scalar equation would be — isolate the unknown matrix, then multiply by a scalar.

Substitute your X back into the original equation and confirm you recover B. It costs one line, it catches a sign error in any entry, and it is the habit the rest of the course rewards.

Multiplication is row against column, and the shapes decide first

Q3 is the definition of the product, met in the order you should always meet it: sizes first, entries second. Part (a) asks for the sizes of both products before either is computed, and the two answers are not the same — one pair of matrices, multiplied in the two orders, gives results that are not even the same shape.

The rules that stop holding — the point of the whole set

Q4 is the question this worksheet exists for. Matrix algebra keeps most of the rules of ordinary arithmetic: addition is commutative and associative, multiplication is associative, and it distributes over addition on both sides. Three familiar rules do not survive, and a student who assumes otherwise loses marks for the rest of the course — in row reduction, in the inverse, in determinants, everywhere.

Part (d) then asks you to prove or disprove the expansion of (A + B)². Do it honestly, with distributivity alone and no assumption about order:

The familiar identity replaces AB + BA by 2AB, and that single replacement is where the usual argument assumes commutativity. Take this away as a method rather than as a fact: whenever a manipulation from ordinary algebra feels automatic, redo it keeping every product in the order it was written, and see whether the two versions still agree. Almost every identity students get wrong on this topic is wrong at exactly that step.

The transpose, and the two families it defines

Q5 introduces the operation that turns row i into column i, and with it two of the special matrices the devis names. Part (a) is the transpose itself together with its size — note that transposing an m × n matrix gives an n × m one, so a non-square matrix and its transpose can never be added to each other or compared.

Part (b) asks for the product of a matrix with its own transpose, and then for the observation that the result is symmetric. The observation is the point, not the arithmetic: entry (i, j) of that product pairs row i of the matrix against row j, and swapping the two rows changes nothing in a sum of products. The symmetry is forced by the definition, so it holds for every matrix and not just for this one.

Part (d) is a short proof of the same shape, and the first question in the set whose answer is an argument rather than a matrix: which square matrices are symmetric and antisymmetric at once? Compare the two entry conditions and the conclusion arrives in two lines, covering every position rather than only the diagonal. Then check that the matrix you land on really does satisfy both conditions — an argument of this kind shows that nothing else can work, and the verification is what turns that into an answer.

Diagonal and triangular: what the shape buys you

Q6 classifies three matrices and then puts the classification to work. Part (a) has a deliberate overlap in it: diagonal is exactly the case where the upper triangular and lower triangular conditions both hold, so one of the three honestly deserves more than one name. Say why, rather than picking whichever label looks most specific.

Part (b) asks for the products of a diagonal matrix with another matrix in both orders, and then for a description in words of what each one does. That description is the useful thing to carry away: multiplying by a diagonal matrix on one side scales the rows, on the other side it scales the columns. Comparing the two results is the cleanest concrete illustration of the non-commutativity from Q4 — they agree in some positions and differ in others, and working out exactly where is the whole content of the part.

Part (c) squares an upper triangular matrix and then asks for the general statement: a product of two upper triangular matrices is upper triangular. The proof is an index argument, and it is a good first one to learn because it is short and entirely mechanical. Take a position below the diagonal, write the sum that defines that entry of the product, and ask which terms can survive: one factor forces the summation index one way, the other forces it the opposite way, and below the diagonal those two demands cannot be met at once. Every term vanishes, so the entry does.

The synthesis: every square matrix splits in two

Q7 closes the set by putting the transpose, addition and scalar multiplication together in one result. Every square matrix is the sum of a symmetric matrix and an antisymmetric one, and the two pieces are written down directly from the matrix and its transpose.

Part (a) computes the two pieces for a given matrix and part (b) verifies all three claims on it — that S is symmetric, that K is antisymmetric, and that they add back to A. Part (c) then asks for the proof in general and hands you the three rules it needs: the transpose of a sum is the sum of the transposes, a scalar comes out of a transpose unchanged, and transposing twice returns the original matrix.

Getting the most out of it

Check the sizes before you compute anything

Sizes settle more questions on this sheet than arithmetic does. Whether a sum exists, whether a product exists, what shape the answer has, whether a matrix is even a candidate for symmetric or triangular — all of it is decided by counting rows and columns, before a single entry is multiplied. Write the sizes above the matrices as you read the question.

Keep every product in the order it was written

The most expensive habit carried over from ordinary algebra is swapping two factors to tidy an expression. AB and BA are different objects. Expand a bracket strictly left to right and leave AB + BA as it stands; and when you feel the urge to cancel a common factor, stop, because that step does not exist here.

Compute the entry you were asked for, and nothing else

A single entry of a product needs one row and one column. Students routinely compute a whole product to report one number, which costs time on an exam and adds a handful of fresh chances to slip. The same economy works as a check: verifying one well-chosen off-diagonal entry usually finds an error faster than recomputing the lot.

Argue from the definition, not from the example

Several parts here ask why — why every antisymmetric matrix carries zeros on its diagonal, why a product of upper triangular matrices stays upper triangular, why a size mismatch makes an operation impossible. In each case the answer is the defining condition written in indices and then read. A worked example is evidence; the index argument is the proof, and 0M04 asks for rigorous mathematical reasoning by name.

Substitute your answer back

Matrix arithmetic has many entries and therefore many places for a sign to go astray, and nearly every question here can be checked by substitution: put the X you solved for back into the equation, add the two pieces of the decomposition and see the original reappear, transpose your result and see that it is what you claimed it was.

Common questions