Determinants · Definition
Cofactor and Laplace expansion
Before you start you need to know
Cofactors reduce a large determinant to smaller ones, and Laplace expansion turns that idea into a computing method.
Definition
The minor \(M\) of an entry is the determinant of the submatrix left after deleting its row and column; the cofactor \(C\) is that minor with an alternating sign.
The formula \(ad-bc\) does not extend obviously to larger matrices. The way out is recursive: pick a row, and multiply each entry by the determinant of what remains after crossing out its row and column (the minor), with an alternating sign. In this way an \(n\times n\) determinant is written using \((n-1)\times(n-1)\) determinants, and the process repeats until size \(2\times2\) is reached.
The sign \((-1)^{i+j}\) follows a checkerboard pattern, starting with \(+\) in the upper left corner. The remarkable fact is that the result does not depend on the chosen row or column, so it is always best to choose the one with the most zeros: each zero removes a whole term.
Guiding example: for \(A=\begin{pmatrix}2&0&1\\3&1&0\\4&0&5\end{pmatrix}\), expanding along the second column leaves a single term, \(1\cdot(+1)\det\begin{pmatrix}2&1\\4&5\end{pmatrix}=6\). Typical mistakes: forgetting the sign of the cofactor, or mixing entries of one row with cofactors of another (that gives zero, a fact used later for the adjugate matrix). For large matrices the method is very expensive, and Gaussian elimination is preferable.
Think about it
What sign does the cofactor of the entry in row 2, column 3 carry?
Negative, because \((-1)^{2+3}=-1\).
Why is it convenient to expand along a row with zeros?
Because each zero entry makes its term \(a_{ij}C_{ij}\) equal to 0, so that minor need not be computed.
Exercises 10
- For \(A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10\end{pmatrix}\), compute the cofactor \(C_{12}\).
- For the same matrix \(A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10\end{pmatrix}\), compute the minor \(M_{23}\) (no sign).
- Which sign \((-1)^{i+j}\) corresponds to position \((3,2)\)?
- Compute \(\det B\) by expanding along the first row, with \(B=\begin{pmatrix}2 & 0 & 1 \\ 3 & 1 & 0 \\ 1 & 2 & 4\end{pmatrix}\).
- Compute the cofactor \(C_{21}\) of \(\begin{pmatrix}2 & 3 \\ 5 & 7\end{pmatrix}\).
- Compute \(\det\begin{pmatrix}1 & 0 & 2 \\ 3 & 0 & 1 \\ 4 & 5 & 6\end{pmatrix}\) by expanding along the most convenient column.
- Compute the determinant of \(\begin{pmatrix}2 & 0 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 4 & 5 & 6 & 0 \\ 7 & 8 & 9 & 1\end{pmatrix}\).
- For \(A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10\end{pmatrix}\), what is \(a_{11}C_{21}+a_{12}C_{22}+a_{13}C_{23}\) (row 1 with the cofactors of row 2)?
- To compute the determinant of a \(4\times4\) matrix by expanding along a row with no zeros, how many \(3\times3\) determinants (minors) must be evaluated?
- Find the values of \(x\) for which \(\det\begin{pmatrix}x & 1 & 0 \\ 2 & x & 1 \\ 0 & 1 & x\end{pmatrix}=0\).
References
- Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §5.2
- Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §10.B