Lecture XXXIII
Determinants; Matrix Algebra
Determinants
For a square matrix
A
, the determinant of
A
has the following properties:
1. Interchanging two rows of the matrix multiplies the value of the determi
nant by 1.
2. If there exists two identical rows, then the value of the determinant is 0.
3. If we multiply a row by a scalar
c
, then the value of the determinant is
also multiplied by
c
.
4. Applying the
β
operation to the matrix leaves the value of the determinant
unchanged.
5. If the determinant is 0, then there exists a row that can be written as a
linear combination of the other rows.
6. Properties (1) through (5) also hold for columns instead of rows.
In addition to the Laplace expansion method of ±nding determinants, there is
a second method that is faster in most cases. This is a simpli±ed
rowreduction
that brings the matrix in to a form called
rowechelon
matrix.
In this method
we use operations
β
and
γ
.
The rowechelon matrix is diﬀerent from the row
reduced matrix in that:
1. pivots can have other values than 1, since we do not use
α
operations;
2. in the column of a pivot, we only need to get 0’s below the pivot.
1
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 Two '04
 Duorg
 Math, Linear Algebra, Determinant, Multiplication, Invertible matrix

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