2: Determinants and Inverses
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Determinants
Consider row reducing the standard 2x2 matrix. Suppose that a is nonzero.
(abcd)
1aR1→R1,R2−cR1→R2
(1bacd)
(1ba0d−cba)
Now notice that we cannot make the lower right corner a 1 if
d−cba=0
or
ad−bc=0.
Definition: The Determinant
We call ad−bc the determinant of the 2 by 2 matrix
(abcd)
it tells us when it is possible to row reduce the matrix and find a solution to the linear system.
Example 2.1
The determinant of the matrix
(3152)
is
3(2)−1(5)=6−5=1.
Determinants of 3 x 3 Matrices
We define the determinant of a triangular matrix
(ade0bf00c)
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 2.2
Find the determinant of
(261024−3042)
We use row operations until the matrix is triangular.
12R1→R1(Multiplies the determinant by 12)
(13524−3042)
R2−2R1→R2 (No effect on the determinant)
(1350−2−13042)
Note that we do not need to zero out the upper middle number. We only need to zero out the bottom left numbers.
R3+2R2→R3 (No effect on the determinant).
(1350−2−1300−24)
Note that we do not need to make the middle number a 1.
The determinant of this matrix is 48. Since this matrix has 12 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
AA−1=A−1A=I.
Example 2.3
Let
A=(2513)
then
A−1=(3−5−12)
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 2.4
(2−11−1)
First note that the determinant of this matrix is
−2+1=−1
hence the inverse exists. Now we set the augmented matrix as
(2−1101−101)
R1↔R2
(1−1012−110)
R2−2R1→R2
(1−101011−2)
R1+R2→R1
(101−1011−2)
Notice that the left hand part is now the identity. The right hand side is the inverse. Hence
A−1=(1−11−2)
Solving Equations Using Matrices
Example 2.5
Suppose we have the system
2x−y=3
x−y=4
Then we can write this in matrix form
Ax=b
where
A=(2−11−1),x=(xy),andb=(34)
We can multiply both sides by A−1:
A−1Ax=A−1b
or
x=A−1b
From before,
A−1=(1−11−2)
Hence our solution is
(−1−5)
or
x=−1 and y=5
Contributors
- Larry Green (Lake Tahoe Community College)
Integrated by Justin Marshall.