Linear Algebra – Determinants

Definition: If \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), then we have: \(\det(A)=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc\).

Definition: If \(A=(a_{ij})\) is an \(m\times n\) matrix, then \(A_{ij}\) denotes the \((m-1)\times(n-1)\) matrix that is obtained from \(A\) by deleting the \(i^{th}\) row and the \(j^{th}\) column. Such a matrix that is obtained from \(A\) by deleting (\(0\) or more) rows and (\(0\) or more) columns is called a submatrix of \(A\).

Definition: If \(A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), then we have

\[\det(A)=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix} =a_{11}\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}-a_{12}\begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix} +a_{13}\begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}=a_{11}\det(A_{11})-a_{12}\det(A_{12})+a_{13}\det(A_{13}).\]

Definition: If \(A=(a_{ij})\) is an \(n\times n\) matrix with \(n\geq2\), then we have:

\[\det(A)=|A|=a_{11}\det(A_{11})-a_{12}\det(A_{12})+\ldots+(-1)^{1+n}a_{1n}\det(A_{1n})=\sum_{j=1}^n(-1)^{1+j}a_{1j}\det(A_{1j})\]

is called the determinant of \(A\).

Definition: If \(A=(a_{ij})\) is an \(n\times n\) matrix with \(n\geq2\), then we have:

\[C_{ij}=(-1)^{i+j}\det(A_{ij})\]

is called a cofactor of \(A\).

Theorem: If \(A=(a_{ij})\) is an \(n\times n\) matrix with \(n\geq2\), then we have:

\[\det(A)=|A|=a_{i1}C_{i1}+a_{i2}C_{i2}+\ldots+a_{in}C_{in}=a_{1j}C_{1j}+a_{2j}C_{2j}+\ldots+a_{nj}C_{nj}\]

for every \(i\in\{1,2,\ldots,n\}\) (the cofactor expansion across the \(i^{th}\) row) and for every \(j\in\{1,2,\ldots,n\}\) (the cofactor expansion across the \(j^{th}\) column).

Theorem: If \(A\) is a suqare matrix, then we have: \(A\) is invertible \(\;\Longleftrightarrow\;\) \(\det(A)\neq0\).

---

Last modified on March 1, 2021