LINEAR ALGEBRA (Math 211)

**Sheet #4**

**DETERMINANTS 2**

**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 ,
then
is multiplied by .

(2)* Interchanging any two rows (or columns) of **A* changes the sign
of .

(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
*characteristic equation*
).
**Definition.** An *eigenvector* corresponding to an eigenvalue
of a matrix *A* is a nontrivial solution of the system

**Definition.** Let *A*=(*a*_{ij}) be a square matrix. The *minor*
*M*_{ij} of
an entry *a*_{ij} is the determinant of the matrix obtained from *A* by
deleting the *i*-th row and *j*-th column. The number
*C*_{ij}=(-1)^{i+j}*M*_{ij} is called the *cofactor* of antry *a*_{ij}.
**Theorem. (Cofactor expansion)**

**Cramer's rule.**

*has a unique solution*