AcademyDeterminants
Academy
General Determinant
Level 1 - Linear Algebra topic page in Determinants.
Principle
The general determinant extends the small determinant formulas to every square matrix. It can be defined by expanding along a row or column, using smaller determinants called minors.
The signs alternate in a checkerboard pattern.
Notation
\(A\)
an n by n matrix
\(a_{ij}\)
entry in row i and column j
\(A_{ij}\)
the matrix obtained by deleting row i and column j from A
\(M_{ij}\)
the minor determinant \det(A_{ij})
\(C_{ij}\)
the cofactor \((-1)^{i+j}M_{ij}\)
The Core Method
The cofactor of entry \(a_{ij}\) is
Cofactor
\[C_{ij}=(-1)^{i+j}\det(A_{ij})\]
Expansion along row \(i\) is
Row expansion
\[\det A=a_{i1}C_{i1}+a_{i2}C_{i2}+\cdots+a_{in}C_{in}\]
Expansion along column \(j\) is
Column expansion
\[\det A=a_{1j}C_{1j}+a_{2j}C_{2j}+\cdots+a_{nj}C_{nj}\]
The sign pattern starts with a plus sign in the top-left corner:
Cofactor signs
\[\begin{pmatrix}+&-&+&\cdots\\-&+&-&\cdots\\+&-&+&\cdots\\\vdots&\vdots&\vdots&\ddots\end{pmatrix}\]
Choose a row or column with many zeros when possible.
Worked Cases
Question
Compute \(\det\begin{pmatrix}1&2&0\\3&-1&4\\2&0&5\end{pmatrix}\) by expanding along the first row.
Answer
Expand along row \(1\): \(\det A=1C_{11}+2C_{12}+0C_{13}\). Compute \(C_{11}=(-1)^2\det\begin{pmatrix}-1&4\\0&5\end{pmatrix}=(-1)(5)-4(0)=-5\). Compute \(C_{12}=(-1)^3\det\begin{pmatrix}3&4\\2&5\end{pmatrix}=-(3\cdot5-4\cdot2)=-(15-8)=-7\). Therefore \(\det A=1(-5)+2(-7)+0=-19\).
Examples
Question
What is the sign attached to the minor in position \((2,3)\)?
Answer
The cofactor sign is \((-1)^{i+j}\). For \((2,3)\), this is \((-1)^{2+3}=(-1)^5=-1\). So \(C_{23}=-M_{23}\).