One scalar that decides invertibility, scales volume, and behaves perfectly under products. The definition takes some care; after that, three row-operation rules do all the computing, and $\det(AB)=\det A\,\det B$ does all the theory.
The determinant is the unique scalar that measures how a linear map scales signed volume — which is why it vanishes exactly when the map squashes space flat, i.e. fails to be invertible.
Jacobian determinants govern change of variables in multivariable calculus and probability; determinants test invertibility in computer algebra; and the characteristic polynomial of Unit 10 is a determinant.
Determinants predate matrices — Seki and Leibniz used them in the 1680s to solve systems. Cauchy proved the product rule in 1812; the modern "multilinear alternating" viewpoint is what survives into higher mathematics.
$\det\!\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)=ad-bc$. For $n\times n$, define recursively by cofactor expansion along the first row: $\det A=\sum_j(-1)^{1+j}a_{1j}\det(A_{1j})$, where $A_{1j}$ deletes row $1$ and column $j$. (Expansion along any row or column gives the same value.)
$A=\left(\begin{smallmatrix}1&2&3\\4&5&6\\7&8&10\end{smallmatrix}\right)$:
$\det$ is linear in each row separately (multilinear), flips sign when two rows are swapped (alternating), and $\det I=1$. These three properties characterize it uniquely. Consequences: a repeated row gives determinant $0$; a triangular matrix's determinant is the product of its diagonal entries; $\det(A^t)=\det A$.
Swapping two rows multiplies $\det$ by $-1$; scaling a row by $c$ multiplies it by $c$; adding a multiple of one row to another leaves it unchanged.
$A$ is invertible iff $\det A\ne0$.
$\det(AB)=\det A\,\det B$. Consequences: $\det(A^{-1})=\dfrac{1}{\det A}$, similar matrices have equal determinants (Unit 6), and $\det(cA)=c^n\det A$ for $n\times n$ matrices (scale each of the $n$ rows).
• $\det(A+B)\ne\det A+\det B$ in general — the determinant is linear in
each row, not in the matrix.
• $\det(cA)=c^n\det A$, not $c\det A$ — every row picks up the factor.
• Track your row operations: forgetting a swap's sign or a scaling factor is
the classic computational error.
• The cofactor signs follow the checkerboard $(-1)^{i+j}$, starting with $+$
in the top-left.
Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.
Compute $\det\!\left(\begin{smallmatrix}2&1\\4&3\end{smallmatrix}\right)$.
Compute $\det\!\left(\begin{smallmatrix}1&2&3\\4&5&6\\7&8&10\end{smallmatrix}\right)$.
If $A$ is $3\times3$ with $\det A=5$, find $\det(2A)$.
Prove $\det(A^{-1})=\dfrac{1}{\det A}$ for invertible $A$.
Prove that a matrix with two equal rows has determinant $0$ (assume the field does not have $1+1=0$).
Show the determinant of an upper-triangular matrix is the product of its diagonal entries.
Find the determinants of the three kinds of elementary matrices.
Compute $\det\!\left(\begin{smallmatrix}1&a\\1&b\end{smallmatrix}\right)$ and interpret the answer for independence of $(1,a)$ and $(1,b)$.
Course coverage. This unit is part of the complete MAT240H1 topic map. It should cover the following items before you move on:
Use the current UofT MAT240H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The determinant spine is: define by universal properties or expansion, prove row-operation rules, then derive multiplicativity and invertibility tests.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
What happens to determinant when two rows are swapped?
Find the determinant of an upper triangular matrix.
State the multiplicativity property.
Why does det$(A) e0$ imply $A$ is invertible?