LINEAR ALGEBRA (Math 211)
Transpose of a matrix.
Product and inverse.
Theorem. A matrix A is invertible if and only if .
Theorem. (Evaluating determinants by row reduction)
(1) If a row (or a column) of A is multiplied by a constant ,
is multiplied by .
(2) Interchanging any two rows (or columns) of A changes the sign
(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
such that the matrix
is not invertible (roots of the
Definition. An eigenvector corresponding to an eigenvalue
of a matrix A is a nontrivial solution of the system
Definition. Let A=(aij) be a square matrix. The minor
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)
has a unique solution