<jupyter_start><jupyter_text>Section 3.3 $\quad$ Cofactor Expansion  Definition of Minor Let $A = [a_{ij}]$ be an $n\times n$ matrix.   Definition of Cofactor Let $A = [a_{ij}]$ be an $n\times n$ matrix.   Example 1 Find the cofactors $A_{12}$ and $A_{23}$ if\begin{equation*}  A =  \left[    \begin{array}{ccc}      4 & 3 & 2 \\      4 & -2 & 5 \\      2 & 4 & 6 \\    \end{array}  \right]\end{equation*}<jupyter_code>from sympy import *

A = Matrix([[4, 3, 2], [4, -2, 5], [2, 4, 6]]);

A.cofactor(0, 1), A.cofactor(1, 2)<jupyter_output><empty_output><jupyter_text>>**Theorem (cofactor expansion)** Let $A = [a_{ij}]$ be an $n\times n$ matrix. Then\begin{equation*}det(A) = \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\end{equation*}and\begin{equation*}det(A) = \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\end{equation*}  Example 2 Find the determinant of\begin{equation*}  A = \left[    \begin{array}{cccc}      1 & 2 & -3 & 4 \\      -4 & 2 & 1 & 3 \\      3 & 0 & 0 & -3 \\      2 & 0 & -2 & 3 \\    \end{array}  \right]\end{equation*}<jupyter_code>from sympy import *

A = Matrix([[1, 2, -3, 4], [-4, 2, 1, 3], [3, 0, 0, -3], [2, 0, -2, 3]]);

A.det()<jupyter_output><empty_output><jupyter_text>Example 3 Find all values of $t$ for which\begin{equation*}  det\left[    \begin{array}{ccc}      t-1 & 0 & 1 \\      2 & t+2 & -1 \\      0 & 0 & t+1 \\    \end{array}  \right] = 0\end{equation*}<jupyter_code>from sympy import *

t = symbols('t');
A = Matrix([[t-1, 0, 1], [2, t+2, -1], [0, 0, t+1]]);

solve(A.det())<jupyter_output><empty_output>
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