| Surviving College Algebra |
| "When all you want is the grade" |
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| Inverses of Matrices |
The
inverse of a matrix is the matrix needed to multiply by to get the identity
matrix. Denote the inverse matrix of
matrix A as A-1. Here is an
equation that explains what the inverse does,
A· A-1 = In
In
order to find an inverse, one must first be able to find determinants. Determinants can only be found in square
matrices, and every square matrix will have a determinant.
To
find the determinant of a 2x2 matrix, first look at a 2x2 matrix.
There
is a pattern to finding the determinant of this matrix. Start at the top left and draw an arrow to
the bottom right.
Multiply
these two numbers (a and d). Draw another arrow from the bottom left to
the top right.
Multiply
these(c and b) and subtract them from the other two (a and
d).
Here
is a summary of the formula for 2x2 matrices.
For
the matrix
the determinant will
equal ad – cb.
Find
the determinant
Set
up the ad-bc equation
2(3)
– 1(4)
Multiply.
6-4
Subtract.
2
Answer!
Some
books, teachers, etc… may use some different symbolism here. If it looks like a matrix but instead of the
brackets has straight lines, then find the determinant. Here is an example of that.
Find
Set
up the equation ad-cb.
1(-5)
- 4(-3)
Multiply.
-5
– (-12)
Two
negatives make a positive.
-5
+ 12
Add.
7
Answer!
A determinant
of a 2x2 matrix is relatively simple to calculate as just seen. In a 3x3 matrix
it will get a little bit more complicated.
Use the arrow method about to be explained, as it is the easiest way to
do these.
Find
the determinant of
The
first thing to do is to copy the matrix so there is one just like it beside it.
Note-
The
reason why the letter i was skipped at the last
element, is because it is easy to get confused with other numbers as being a
1. It also closely resembles a j, many
textbooks, teachers, etc.. will
not use i as a variable for those reasons.
Use
the first matrix (one on the left) as the original. Start at the top left corner and draw a line to the bottom
right.
Just
like in a 2x2, multiply the elements a,e,
and j. The next step is to go to the b
element and draw a diagonal line just like the previous one.
It
is obvious after going through the f that one somewhat runs out of the matrix
that is why the second matrix was copied earlier. Multiply these elements and
add them to the result of the first arrow. Here are the results so far.
aej + bfg.
Moving along to the next element, c, repeat the same process.
Add
the result of this to the previous step.
aej + bfg + cdh.
This
is the halfway mark of the formula, move to the g element next and draw an
arrow to the upper right corner.
With
all of the next arrows, subtract them in the formula.
aej + bfg
+ cdh- gec
Move
on to the next element, h, and draw the arrow,
Subtract
the result of this in the equation.
aej + bfg + cdh- gec- hfa
Moving right along, sart at the j and draw
the arrow up. This is the last step so the equation will be complete.
Plug
in the result making sure to subtract.
aej + bfg
+ cdh- gec- hfa – jdh.
Those
instructions probably seemed very lengthy and possibly difficult to remember,
here is a summary that will help. The
first picture is to state the order in which the arrows need to be drawn.
1 2
3
Then
4 5
6
That
is the order, the next picture shows whether to add or subtract the arrows
+
+ +
And
- - -
Therefore,
the determinant of
is
aej + bfg
+ cdh – gec – hfa – jdb
Example.
Find
the determinant of
Copy
the matrix and draw the arrows going down.
Add
the result of these.
Note-
Anytime
plugging values into a formula, if the value has a negative sign, make sure and
use parentheses around that value. If
not, it is very easy to make a simple sign mistake that will make the answer
wrong!
1(-2)(-1) + 4(0)(-5) + 3(2)(5)
This
is only half of the formula. Draw the arrows going up.
Take
the result of these and they are subtracted in the formula.
1(-2)(-1) + 4(0)(-5) + 3(2)(5) – (-5)(-2)(3) – 5(0)(1) –
(-1)(2)(4)
Multiply
each group of numbers. The ones that
contain a zero are easy because zero times anything will equal zero. Use this
knowledge to save time.
2 +
0 + 30 –30 – 0 – (-8)
Two
negatives make a positive (last term).
2 +
0 + 30 – 30 – 0 + 8
Add
and subtract from left to right.
10
Answer!
Example
Find
Draw
the first arrows, those that go down, and write the result in an equation.
(-2)(2)(1)
+ (-1)(3)(0) + 4(5)(-5)
Draw
the arrows that go up and plug the results into the formula.
(-2)(2)(1)
+ (-1)(3)(0) + 4(5)(-5)- 0(2)(4) – (-5)(3)(-2) –
1(5)(-1)
Multiply
each group of numbers.
- 4
+ 0 + (-100) – 0 – 30 – (-5)
Simplify.
-4
+ 0 –100 – 0 –30 + 5
Add
and subtract from left to right.
-129
Answer!
Remember
that the inverse matrix A is denoted by A-1. Finding the inverse of a 2x2 matrix is done
with the following formula
If 
Then 
The
first part of the formula, 
, is read as “one over the determinant of A.” After
finding this value use it as a scalar and multiply it by each element in the matrix 
.
Example
Let
Find A-1
Find
the determinant, or det, of A.
3(4)
– 2(5)
2
This
is the determinant.
Plug
the 2 into the formula for the determinant.
Multiply
by each
element.
Simply each element that will simplify.
Answer!
Example
Let
Find A-1
Find
the determinant.
(-1)(7)
– (3)(-2)
-7
+ 6
-1
-1
is the determinant. Plug it into the formula.
The
scalar, 
, can be simplified to –1.
Multiply
the scalar, -1.
Answer!
Example
Let
Find
X-1
Find
the determinant.
(-1)(6)
– 3(-2)
-6
+ 6
0
This
is the determinant, plug into the formula.
Look
at the scalar, 
, trying to calculate this one realizes that nothing
can be divided by 0. The result is
undefined. Therefore, the inverse does not exist when the determinant is equal
to 0.
Undefined
Answer!
Inverses
are just one area to apply the determinant, another
type of problem using determinants is Cramer’s Rule. Cramer’s Rule can be used to solve
systems. The easiest way to solve
systems is through row elimination described in that section. But since some
instructors may demand upon the use or knowledge of Cramer’s Rule it will be
explained here.
A
system is a group of equations involving more than one variable. The ultimate goal of solving a system is to
find the value of each variable.
Here
is an example.
2x
+ 3y = 8
-3x
– 4y = -11
To
apply Cramer’s rule make a matrix with the coefficients (the numbers in front
of the letters, x and y). Make this
matrix just like it appears in the system, call this matrix A.
Notice
the 8 and –11 were not used. They will
be used in creating two other matrices later. Create a matrix that will be used
to solve for x. This will be called the
matrix X, used to solve for x.
To
find X, start with the matrix already created, , replace the first column (the
numbers that were in front of x) and replace them with the 8 and –11 in proper
order (as they appear in the problem).
The
formula for the value of x is 
. Therefore,
find det X and det A.
det X or 
= 1
det A or 
= 1
So
the value of x is 
=
=1. x = 1
Next
find y. Create matrix Y the same way matrix X was created only replacing the
coefficients of y with 8 and –11.
The
value of y is 
. Find det Y (det A is 1 from previous
work).
det Y or 
= 2
The
value of y is = 
= 2.
Write
the answer of a system in order, in parentheses. In this case use alphabetical order since it
is letters (x and y).
(1,
2) Would be stating that x =1 and y = 2.
Example
Solve
using Cramer’s Rule
3a1
– a2 = -10
a1
+ a2 = 2
The
two variables to solve for are a1 and a2. So the matrices to create will be called A, A1,
and A2, respectively.
Go
ahead and find the determinant of these matrices
det A = 4
det A1 = -8
det A2 = 16
Value of a1 is 
= 
= -2
Value of a2 is 
= 
= 4
Write the result in the proper order. Since the two
letters are the same, write them in the order of their name, a1 and
then a2.
(- 2 , 4)
Answer!