Determinant of a Square Matrix (2×2 and 3×3)

What Is a Determinant?

Associated with every square matrix $A$ is a unique number called its determinant, written $|A|$ or $\det A$. Unlike a matrix (an array of numbers), a determinant is a single scalar value computed from the entries of the matrix according to a fixed rule. Determinants first arose historically from the problem of solving simultaneous linear equations -- indeed the word itself reflects that a determinant "determines" whether a system of linear equations has a unique solution (Section 7). This chapter develops determinants only for square matrices of order up to $3\times3$, matching the matrices met so far in this course.

For a matrix of order $1\times1$, $A=[a]$, the determinant is simply the entry itself: $|A|=a$.

Determinant of a $2\times2$ Matrix

For $A=\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}$, the determinant is defined as

that is, the product of the entries on the leading (top-left to bottom-right) diagonal minus the product of the entries on the other diagonal. For example, $\begin{vmatrix}5&3\\2&4\end{vmatrix}=5(4)-3(2)=20-6=14$.

This rule is not arbitrary -- it falls straight out of solving a general pair of linear equations $a_{11}x+a_{12}y=b_1,\ a_{21}x+a_{22}y=b_2$ by elimination: multiplying the first equation by $a_{22}$ and the second by $a_{12}$ and subtracting eliminates $y$, and the coefficient of $x$ that survives is exactly $a_{11}a_{22}-a_{12}a_{21}$ (Section 8 makes this connection explicit).

Determinant of a $3\times3$ Matrix

For $A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$, the determinant is defined by expansion along the first row, reducing the $3\times3$ determinant to a combination of three $2\times2$ determinants:

Each $2\times2$ determinant on the right is obtained by deleting the row and column of the entry it is multiplied by -- e.g. the first $2\times2$ determinant is what remains after deleting row 1 and column 1 (the row and column of $a_{11}$). The alternating $+,-,+$ signs in front of the three terms follow a fixed checkerboard pattern that is developed fully as cofactors in Section 3.

Illustration. For $A=\begin{pmatrix}1&2&3\\0&1&4\\2&1&0\end{pmatrix}$:

Expansion Along Any Row or Column

Although the determinant was defined using the first row, it is a remarkable and extremely useful fact -- proved in Section 3 once cofactors are introduced -- that expanding along any row or any column of $A$ gives exactly the same numerical value. This freedom is one of the most practically important ideas in the whole chapter: a row or column that already contains one or more zeros should always be chosen for expansion, since every zero entry eliminates the need to compute the $2\times2$ determinant it would otherwise multiply.