
Determinants:
Consider row reducing the standard 2x2 matrix. Suppose that a is nonzero.
1/a R_{1} > R_{1}

R_{2}  cR_{1} > R_{2}



Now notice that we cannot make the lower right corner a 1 if
d  cb/a = 0
or
ad  bc = 0
Definition of the Determinant
We call ad  bc the determinant of the 2 by 2
matrix 

it tells us when it is possible to row reduce the matrix and find a solution
to the linear system.
Example:
The determinant of the matrix
is
3(2)  1(5) = 6  5 = 1

Determinants of Three by Three Matrices
We define the determinant of a triangular matrix
by
det = abc
Notice that if we multiply a row by a constant k then the new determinant
is k times the old one. We list the effect of all three row operations
below.
Theorem
The effect of the the three basic row operations on
the determinant are as follows
 Multiplication of a row by a constant multiplies the determinant
by that constant.
 Switching two rows changes the sign of the determinant.
 Replacing one row by that row + a multiply of another row has no
effect on the determinant.

To find the determinant of a matrix we use the operations to make the matrix
triangular and then work backwards.
Example:
Find the determinant of
We use row operations until the matrix is triangular.
1/2 R_{1} <> R_{1}
(Multiplies the determinant by
1/2)
R_{2}  2R_{1} > R_{2}
(No effect
on the determinant)
Note that we do not need to zero out the upper
middle number. We only need to zero out the bottom left numbers.
R_{3} + 2R_{2} > R_{3}
(No effect on the determinant)
Note that we do not need to make the middle number
a 1.
The determinant of this matrix is 48. Since this matrix has 1/2 the determinant
of the original matrix, the determinant of the original matrix has
determinant = 48(2) = 96.

Inverses
We call the square matrix I with all 1's down the diagonal and zeros
everywhere else the identity matrix. It has the unique property
that if A is a square matrix with the same dimensions then
AI = IA = A
Definition
If A is a square matrix then the inverse A^{1}
of A is the
unique matrix such that
Example:
Let
then
Verify this!
Theorem
The inverse of a matrix exists if and only if the
determinant is nonzero. 
To find the inverse of a matrix, we write a new extended matrix with the
identity on the right. Then we completely row reduce, the resulting
matrix on the right will be the inverse matrix.
Example:
First note that the determinant of this matrix is
2 + 1 = 1
hence the
inverse exists. Now we set the augmented matrix as

R_{1} <> R_{2} 
R_{2}  2R_{1} > R_{2}

R_{1} + R_{2} > R_{1}





Notice that the left hand part is now the identity. The right hand
side is the inverse. Hence

Solving Equations Using Matrices
Example:
Suppose we have the system
2x  y = 3
x  y = 4
Then we can write this in matrix form
Ax = b
where
We can multiply both sides by A^{1}:
A^{1}A x = A^{1}b
or
x = A^{1}b
From before,
Hence our solution is
or
x = 1
and y = 5

The Easy Way
A graphing calculator can be used to work all of the above problems.