Matrix Determinant

Compute Matrix Determinant
A square matrix \(A\), \[\begin{aligned} A = \begin{bmatrix} a_{1,1} & a_{1,2} & ... & a_{1,n} \\ a_{2,1} & a_{2,2} & ... & a_{2,n} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n,1} & a_{n,2} & \ldots & a_{n,n} \end{bmatrix} \end{aligned}\] Be \(A\) a matrix \(n \times n\) and \(n \geqslant 2\). The determinant of \(A\) is scalar give by \(|A| = det(A)\), \[det(A) = \sum_{k=1}^{n} a_{i,j} C_{i,j}\] where \(C_{i,j}\) is cofactor, \[C_{i,j} = (-1)^{i+j} det( A_{i,j} )\] Example \(\bullet\) Compute the determinant of matrix \(A\) showed bellow, \[\begin{aligned} A = \begin{bmatrix} 1 & 1 & 3 \\ 2 & 5 & 7 \\ 12 & 6 & 1 \end{bmatrix} \end{aligned}\] Resolution Choice expansion for row \(1\), \[det(A) = \sum_{k=1}^{3} a_{1,k} (-1)^{1+k} det\left( A_{1,k} \right) = \]
\[ = 1 (-1)^{1+1 }det \left( \begin{bmatrix} 5 & 7 \\ 6 & 1 \end{bmatrix} \right) + 1 (-1)^{1+2} det \left( \begin{bmatrix} 2 & 7 \\ 12 & 1 \end{bmatrix} \right) + 3 (-1)^{1+3} det \left( \begin{bmatrix} 2 & 5 \\ 12 & 6 \end{bmatrix} \right) = \]
\[= 1 \cdot ( 5 \cdot 1 - 7 \cdot 6 ) + (-1)( 2 \cdot 1 - 7 \cdot 12 ) + 3 \cdot 1 \cdot (2 \cdot 6 - 5 \cdot 12 ) = \\ \newline \]
\[= -37 + 82 - 144 = \boxed{-99}\]

PDF Text: Determinant of Matrix A