2: Determinants and Inverses
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Determinants
Consider row reducing the standard 2x2 matrix. Suppose that
Now notice that we cannot make the lower right corner a 1 if
or
Definition: The Determinant
We call
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
Determinants of 3 x 3 Matrices
We define the determinant of a triangular matrix
by
Notice that if we multiply a row by a constant
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.
Note that we do not need to zero out the upper middle number. We only need to zero out the bottom left numbers.
Note that we do not need to make the middle number a 1.
The determinant of this matrix is 48. Since this matrix has
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
Definition
If
Example
Let
then
Verify this!
Theorem: ExistEnce
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
hence the inverse exists. Now we set the augmented matrix as
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
Then we can write this in matrix form
where
We can multiply both sides by
or
From before,
Hence our solution is
or
Contributors
- Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.