5. Finding the Inverse of a Matrix
by M. Bourne
What are we doing?
If we multiply matrix A by the inverse of matrix A, we will get the identity matrix, I.
The concept of solving systems using matrices is similar to the concept of solving simple equations.
For example, to solve 7x = 14, we multiply both sides by the same number. We find the "inverse" of `7`, which is `1/7`. Multiplying both sides on the left by `1/7` gives:
`(1/7) × 7x = (1/7) × 14`
On the left, we have (in part):
`(1/7) × 7 = 1`
The number `1` is the "identity" for multiplication of ordinary numbers. On the right, we get:
`1/7 xx 14 = 2`.
The solution for our equation is:
x = 2
We extend this concept of finding an inverse for solving a single equation, to solving systems of simultaneous equations.
We need to find inverses of matrices so that we can solve systems of simultaneous equations.
(We'll see how to solve systems in the next section, Matrices and Linear Equations).
We'll find the inverse of a matrix using 2 different methods. You can decide which one to use depending on the situation.
The first method is limited to finding the inverse of 2 × 2 matrices. It involves the use of the determinant of a matrix which we saw earlier.
Reminder: We can only find the determinant of a square matrix. For example, if A is the square matrix
`((2,3),(-1,5))`
then we can find the determinant of A:
`|(2,3),(-1,5)|=10+3=13`
For convenience, we could have written the determinant of matrix `A` as `|A|` and so our final answer would be:
`|A| = 13`
Another way of writing the same thing is to use "det" for "determinant". So for example, in this case we would write:
`det(A) = 13`
Method 1 - Transposing and Determinants
This method is only good for finding the inverse of a 2 × 2 matrix.
We'll see how this method works via an example.
Example
Find the inverse, `A^-1`, of
`A=((2,-3),(4,-7))`
using Method 1.
Answer
Method 1 is as follows.
[1] Interchange leading diagonal elements:
`-7 → 2`; `2 → -7`
`((-7,-3),(4,2))`
[2] Change signs of the other 2 elements:
`-3 → 3`; `4 → -4`
`((-7,3),(-4,2))`
[3] Find the determinant `|A|`
`|(2,-3),(4,-7)|=-14+12=-2`
[4] Multiply result of [2] by `1/|A|`
`A^-1 = 1/|A|((-7,3),(-4,2))`
`=1/(-2)((-7,3),(-4,2))`
`=((3.5,-1.5),(2,-1))`
So we have found the inverse, as required.
Is it correct?
We check by multiplying our inverse by the original matrix. If we get the identity matrix (I) for our answer, then we must have the correct answer.
`A^-1A=((3.5,-1.5),(2,-1))((2,-3),(4,-7))`
`=((7-6,-10.5+10.5),(4-4,-6+7))`
`=((1,0),(0,1))`
`=I`
We can go to bed happy, knowing that our answer is correct.
Method 2 - Adjunct Matrix (can be extended to any size)
NOTE: I have left Method 2 here for historical reasons. We will be using computers to find the inverse (or more importantly, the solution for the system of equations) of matrices larger than 2×2.
If you need to find the inverse of a 3×3 (or bigger) matrix using paper, then follow the steps given. It is tedious, but it will get you there. Good luck.
Method 2 uses the adjoint matrix method.
[Warning: This is long - and ancient history!]
Answer
Method 2 (an example of dinosaur mathematics - should be extinct)
The inverse of a 3×3 matrix is given by:
`A^-1=("adj"A)/(detA)`
"adj A" is short for "the adjoint of A". We use cofactors (that we met earlier) to determine the adjoint of a matrix.
Cofactors
Recall: The cofactor of an element in a matrix is the value obtained by evaluating the determinant formed by the elements not in that particular row or column.
Example 2a
Consider the matrix:
`((5,6,1),(0,3,-3),(4,-7,2))`
The cofactor of 6 is
`|(0,-3),(4,2)|=0+12=12`
The cofactor of -3 is
`|(5,6),(4,-7)|=-35-24=-59`
We find the adjoint matrix by replacing each element in the matrix with its cofactor and applying a + or - sign as follows:
`((+,-,+),(-,+,-),(+,-,+))`
and then finding the transpose of the resulting matrix. The transpose means the 1st column becomes the 1st row; 2nd column becomes 2nd row, etc.
Example 2b
Find the inverse of the following by using the adjoint matrix method:
`A=((5,6,1),(0,3,-3),(4,-7,2))`
Solution
Step 1:
Replace elements with cofactors and apply + and -
`((+(-15),-(12),+(-12)),(-(19),+(6),-(-59)),(+(-21),-(-15),+(15)))`
`=((-15,-12,-12),(-19,6,59),(-21,15,15))`
Step 2
Transpose the matrix:
`"adj"A = ((-15,-19,-21),(-12,6,15),(-12,59,15))`
Before we can find the inverse of matrix A, we need det A:
`|(5,6,1),(0,3,-3),(4,-7,2)|` `=5(-15)+4(-21)` `=-159`
Now we have what we need to apply the formula
`A^-1=("adj"A)/detA`
So
`A^-1=("adj"A)/detA`
`=1/-159((-15,-19,-21),(-12,6,15),(-12,59,15))`
`A^-1=((0.094,0.119,0.132),(0.075,-0.038,-0.094),(0.075,-0.371,-0.094))`
Example 2c
Find the inverse of
`((-2,6,1),(0,3,-3),(4,-7,3))`
using Method 2.
Solution
`text(C of) A` `=((+(-12),-(12),+(-12)),(-(25),+(-10),-(-10)),(+(-21),-(6),+(-6)))`
`=((-12,-12,-12),(-25,-10,10),(-21,-6,-6))`
Interchange rows and columns:
`"adj"A=((-12,-25,-21),(-12,-10,-6),(-12,10,-6))`
`"det"A`
`=|(-2,6,1),(0,3,-3),(4,-7,3)|`
`=2(9-21)+4(-21)`
`=-60`
So
`A^-1=("adj"A)/(detA)`
`=1/-60((-12,-25,-21),(-12,-10,-6),(-12,10,-6))`
`=( (1/5,5/12,7/20),(1/5,1/6,1/10),(1/5,-1/6,1/10))`
`=((0.2,0.417,0.35),(0.2,0.167,0.1),(0.2,-0.167,0.1))`
Using a computer to find matrix inverses
Now let's see how to do all this more appropriately using a computer...
Inverses of Larger Matrices (Method 3)
Most real systems of equations are very large (up to 100 by 100 is common). We use computers to find these inverses. You need to understand what to give the computer and what it will give you as an answer.
However, some people need to know how to find inverses of large matrices!
See Inverse of a Matrix Using Gauss-Jordan Elimination for the most common method for finding inverses.
Exercise
Find the inverse of
`((7,-2),(-6,2))`
by Method 1.
(I believe this is the level of inverse we should do on paper, so we get a sense of what an inverse is and how it may be calculated. Anything bigger than this should be done using computer :-)
Answer
[1] Interchange leading diagonal elements:
`((2,-2),(-6,7))`
[2] Change signs of the other 2 elements:
`((2,2),(6,7))`
[3] Find |A|
Remember that our original matrix (from the question) is
`A=((7,-2),(-6,2))`
So the determinant of A is given by:
`|A|=|(7,-2),(-6,2)|=14-12=2`
[4] Multiply result of [2] by `1/|A|`
`A^-1=1/(|A|)((2,2),(6,7))`
`=1/2((2,2),(6,7))`
`=((1,1),(3,3.5))`
Is it correct?
Check:
`A^-1A=((1,1),(3,3.5))((7,-2),(-6,2))`
`=((7-6,-2+2),(21-21,-6+7))`
`=((1,0),(0,1))`
`=I`
Let's now see some examples of products and inverses of matrices.