LINEAR ALGEBRA (Math 211)
Sheet #4
DETERMINANTS 2

Transpose of a matrix.

\begin{displaymath}\mbox{\scriptsize $\left(\begin{array}{ccc} a_{11}&\dots&a_{1... ...dot A^T$}\vspace{20pt}\\ \fbox{$\det(A^T)=\det A$}\end{array}\end{displaymath}



Product and inverse.
Theorem. A matrix A is invertible if and only if $\det A\not= 0$.

\begin{displaymath}\fbox{$\det({\alpha}\cdot A)={\alpha}^n\det A$}\qquad\qquad ... ...fbox{$\displaystyle \det(A^{-1})={1\over\det A}$}\vspace{-5pt} \end{displaymath}

Theorem. (Evaluating determinants by row reduction)
(1) If a row (or a column) of A is multiplied by a constant ${\alpha}$, then $\det A$ is multiplied by ${\alpha}$.
(2) Interchanging any two rows (or columns) of A changes the sign of $\det A$.
(3) If a multiple of one row (column) of A is added to another row (column) of A, then the determinant is unchanged.



Eigenvalues and eigenvectors
Definition. An eigenvalue of a matrix A is a value of ${\lambda}$ such that the matrix $A-{\lambda}I$ is not invertible (roots of the characteristic equation $\det(A-{\lambda}I)=0$).

Definition. An eigenvector corresponding to an eigenvalue ${\lambda}$ of a matrix A is a nontrivial solution of the system $A\cdot X = {\lambda}\cdot X$



Definition. Let A=(aij) be a square matrix. The minor Mij of an entry aij is the determinant of the matrix obtained from A by deleting the i-th row and j-th column. The number Cij=(-1)i+jMij is called the cofactor of antry aij.

Theorem. (Cofactor expansion)

$\displaystyle \begin{array}{c} \det(A) = a_{1j}C_{1j}+a_{2j}C_{2j}+\dots+a_{nj}C_{nj}\\ \mbox{(cofactor expansion along the $j$-th column)} \end{array}$                  $\displaystyle \begin{array}{c} \det(A) = a_{i1}C_{i1}+a_{i2}C_{i2}+\dots+a_{in}C_{in}\\ \mbox{(cofactor expansion along the $i$-th row)} \end{array}$


$\textstyle \parbox{7.5cm}{{\bf Theorem. (Inverse matrix)}\\ \hspace*{\fill}$\d... ... \vdots&\vdots& &\vdots\\ C_{1n}&C_{2n}&\dots&C_{nn} \end{array}\right)$}\ .$}$ $\textstyle \parbox{9cm}{\fbox{\begin{tabular}{l} {\bf Vandermonde determinant}\... ...\ \end{array}\right\vert$} = \prod_{1\leq j<i\leq n} (x_i-x_j)$\end{tabular}}}$



Cramer's rule.

$\textstyle \parbox{6.5cm}{{\it If $\det A\not= 0$\ then the system of linear equations}}$      $\left\{\begin{array}{ccccccccc} a_{11}x_1&+&a_{12}x_2&+&\dots&+&a_{1n}x_n&=&b_1... ... &\vdots \\ a_{n1}x_1&+&a_{n2}x_2&+&\dots&+&a_{nn}x_n&=&b_n \end{array}\right.$ has a unique solution


\begin{displaymath} x_1 = {\mbox{\scriptsize $\left\vert\begin{array}{cccc} b_1&... ... a_{n1}&a_{n2}&\dots&b_n \end{array}\right\vert$} \over \det A}\end{displaymath}