Cofactor expansions

Lecture 21

Author

Affiliation

Minjae Park

Auburn University
MATH 2660 - Spring 2026

Published

February 27, 2026

Recap

Determinant of \(2\times2\) Matrix

Let \(A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\).
We define the determinant of \(A\) by \(\det(A)=\begin{vmatrix} a & b \\ c & d \end{vmatrix}=ad-bc\).
The condition \(\det(A)=0\) holds if and only if the column vectors \(\vec{u}_1\) and \(\vec{u}_2\) are linearly dependent.
The determinant measures the signed area of the parallelogram formed by the columns of \(A\).
The absolute value \(|\det(A)|\) gives the area scale factor, while the sign records the orientation (according to the right-hand rule).

Example

Suppose that an Aubie is drawn whose area is 10. Let \(A(x,y)=(-x-y,-x+y)\) be a linear transformation on the plane. If the Aubie is transformed by \(A\), what is a rough shape and what would be the area of the deformed Aubie?
- Find the matrix representation of \(A\).
- Compute the determinant of \(A\).
- Submit the area answer in iClicker.

Scan the QR code or go to join.iclicker.com/MBNJ.

Determinants for Higher Dimensions

Determinants for \(3\times3\) Matrices

The determinant idea extends naturally to higher dimensions.
Let \(A=[\vec{u}_1\ \vec{u}_2\ \vec{u}_3]\) be a \(3\times3\) matrix with column vectors \(\vec{u}_i\in\mathbb R^3\).
The determinant \(\det(A)\) measures the signed volume of the parallelepiped formed by \(\vec{u}_1,\vec{u}_2,\vec{u}_3\).
If this volume is nonzero, the three vectors form a genuine three-dimensional grid and are linearly independent.
If the volume is zero, the grid collapses into a plane or even a line, so the vectors are linearly dependent.
You can visualize this at Linear Transformations in 3D.
Therefore, \(\det(A)\) provides a test for linear independence in \(\mathbb R^3\).

Idea from the \(2\times2\) Case

Recall the Geometric Proof for \(2\times2\) determinants.
There we “flatten” one vector to simplify the area computation.
Instead of working directly with the parallelogram formed by \(\langle a,c \rangle\) and \(\langle b,d \rangle\), we decompose \(\langle a,c \rangle = \langle a,0 \rangle + \langle 0,c \rangle\).
First consider the parallelogram formed by \(\langle a,0 \rangle\) and \(\langle b,d \rangle\); its signed area is \(ad\).
Then consider the parallelogram formed by \(\langle 0,c \rangle\) and \(\langle b,d \rangle\); its signed area is \(bc\).
Because of orientation, these contributions have opposite signs.
Therefore the total signed area is \(ad-bc\).
Similarly, in higher dimensions, we choose one column vector and split it into pieces where all coordinates are zero except one.

The \(3\times3\) Case: Expand Along Column 1

Let \(A=\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\).
We isolate entries in the first column: \(a\), \(d\), and \(g\).
First term: keep \(a\) and zero out the rest of its column, \(\begin{bmatrix} a & b & c \\ 0 & e & f \\ 0 & h & i \end{bmatrix}\).
Here \(a\) acts like a “height” and the base is formed by \(\begin{bmatrix} e & f \\ h & i \end{bmatrix}\).
Contribution: \(a\begin{vmatrix} e & f \\ h & i \end{vmatrix}\).

Continuing the Column Expansion

Second term: isolate \(d\), \(\begin{bmatrix} 0 & b & c \\ d & e & f \\ 0 & h & i \end{bmatrix}\).
Here \(d\) acts like a “height” and the base is formed by \(\begin{bmatrix} b & c \\ h & i \end{bmatrix}\).
Because of orientation, this term carries a minus sign.
Contribution: \[-\,d\begin{vmatrix} b & c \\ h & i \end{vmatrix}.\]

Continuing the Column Expansion

Third term: isolate \(g\), \(\begin{bmatrix} 0 & b & c \\ 0 & e & f \\ g & h & i \end{bmatrix}\).
Here \(g\) acts like a “height” and the base is formed by \(\begin{bmatrix} b & c \\ e & f \end{bmatrix}\).
Contribution: \[g\begin{vmatrix} b & c \\ e & f \end{vmatrix}.\]

Cofactor Expansion Formula (Column 1)

Collecting all terms, we obtain \[ \det(A) = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} - d\begin{vmatrix} b & c \\ h & i \end{vmatrix} + g\begin{vmatrix} b & c \\ e & f \end{vmatrix}. \]
This is called cofactor expansion along the first column.
We could instead expand along the second or third column (or along any row) and obtain the same determinant.
The signs are determined by the checkerboard pattern \[ \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}. \]

Minors and Cofactors for \(3\times3\) Matrices

Let \(A\) be a \(3\times3\) matrix.
The minor \(M_{ij}\) is the determinant obtained by deleting row \(i\) and column \(j\) from \(A\).
The cofactor corresponding to \(A_{ij}\) is \[C_{ij}=(-1)^{i+j}M_{ij}.\]
Cofactor expansion along column 1 is \[ \det(A)=A_{11}C_{11}+A_{21}C_{21}+A_{31}C_{31}. \]
More generally, cofactor expansion along column \(j\) (where \(j=1,2,3\)) is \[ \det(A)=A_{1j}C_{1j}+A_{2j}C_{2j}+A_{3j}C_{3j}. \]

Example

Compute \[ \det\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}. \]
Expand along the first column: \[ =1\begin{vmatrix}4 & 5 \\ 0 & 6\end{vmatrix} -0\begin{vmatrix}2 & 3 \\ 0 & 6\end{vmatrix} +1\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = 24-0+(-2) = 22. \]
See Cofactor Expansion for 3×3 Determinants for a visual walkthrough. <!– - Compute minors:
- \(\begin{vmatrix}4 & 5 \\ 0 & 6\end{vmatrix}=24\),
- \(\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix}=10-12=-2\).
Final answer: \[ 1(24)+1(-2)=22. \] –>

Determinants for \(n\times n\) Matrices

In general, let \(A=[\vec{u}_1\ \vec{u}_2\ \dots\ \vec{u}_n]\) be an \(n\times n\) matrix.
The determinant of \(A\), denoted \(\det(A)\), is the signed hypervolume of the hyperparallelogram formed by its \(n\) column vectors.
If \(\det(A)=0\), the hyperparallelogram collapses into a lower-dimensional object, so the column vectors are linearly dependent.
If \(\det(A)\neq 0\), the hyperparallelogram has nonzero hypervolume, so the column vectors are linearly independent.
The determinant can be computed recursively using cofactor expansion, extending the \(2\times2\) and \(3\times3\) procedures.

Minors and Cofactors for \(n\times n\) Matrices

The minor \(M_{ij}\) is the determinant obtained by deleting row \(i\) and column \(j\) from \(A\).
The cofactor corresponding to \(A_{ij}\) is \(C_{ij}=(-1)^{i+j}M_{ij}\).
Cofactor expansion along column \(j\) (where \(j=1,2,\dots,n\)) is \[ \det(A)=\sum_{i=1}^n A_{ij}C_{ij} =\sum_{i=1}^n (-1)^{i+j}A_{ij}M_{ij}. \]
Cofactor expansion along row \(i\) (where \(i=1,2,\dots,n\)) is \[ \det(A)=\sum_{j=1}^n A_{ij}C_{ij} =\sum_{j=1}^n (-1)^{i+j}A_{ij}M_{ij}. \]
See Cofactor Expansion for \(4\times4\) Determinants for a visual walkthrough.

Determinants for Special Matrices

Let \(D\) be an \(n\times n\) diagonal matrix with diagonal entries \(d_1,\dots,d_n\). Then \[\det(D)=d_1d_2\cdots d_n.\]
More generally, if \(T\) is a triangular matrix (upper or lower) with diagonal entries \(d_1,\dots,d_n\), then \[\det(T)=d_1d_2\cdots d_n.\]
If \(E_{R_i\leftrightarrow R_j}\) is the elementary matrix corresponding to swapping two rows, then \[\det\!\left(E_{R_i\leftrightarrow R_j}\right)=-1.\]
Geometrically, swapping two rows reverses orientation, which explains the negative sign.

Determinant and Invertible Matrices

Geometrically, \(A^{-1}\) undoes what \(A\) does to vectors in \(\mathbb R^n\).
If the columns of \(A\) are linearly independent, the associated hyperparallelogram has nonzero hypervolume — nothing collapses to a lower dimension.
Since no information is lost, the transformation can be reversed; hence \(A\) is invertible.
From a row-reduction perspective: if the \(n\) columns are linearly independent, then \(\text{rank}(A)=n\), so the RREF of \(A\) has \(n\) pivots and equals \(I_n\).
Therefore \(A\) can be transformed into \(I_n\) by a sequence of elementary matrices, and those same operations produce \(A^{-1}\).
Consequently, \(\det(A)\neq 0 \quad \Longleftrightarrow \quad A \text{ is invertible.}\)

Computing Determinants

Computing large \(n\times n\) determinants by hand is generally impractical; computer algebra systems (e.g., MATLAB or Mathematica) are far more efficient.
Caution: AI tools are not reliable for exact computations!
Nevertheless, you should be comfortable computing several \(2\times2\) and \(3\times3\) determinants by hand to understand the underlying mechanism.
In this course, the emphasis is on understanding what the determinant means, rather than performing lengthy computations.
You will not be asked to compute, for example, a \(4\times4\) determinant by hand on an exam.

What You Need to Know About Determinants

For an \(n\times n\) matrix \(A\):
- \(\det(A)\) can be computed by cofactor expansion along any row or any column; in practice, choose one with many zeros to simplify the computation.
- \(|\det(A)|\) equals the hypervolume of the hyperparallelogram formed by its \(n\) column vectors.
- The sign of \(\det(A)\) records the orientation of the column vectors.
- If \(\det(A)=0\), the columns are linearly dependent.
- If \(\det(A)\neq 0\), the columns are linearly independent and \(A\) is invertible.