Matrices and their inverse?

Question

How do you find the inversr to a matrix?

Answer 1

First consider the definition of an inverse of a matrix. What is it? What does it mean? An inverse matrix is a matrix when multiplied by the original matrix you get the identity matrix. In other words if A is a matrix and A-inverse is a matrix then
A X (A-inverse) = I (*).

So how does this help you? Well, set up an augmented matrix as follows.

Suppose I want to find the inverse of the matrix A =
[1 2 3]
[2 0 6]
[0 9 3]

Then the first thing I should do is set up the following augmented matrix.

[1 2 3 | 1 0 0]
[2 0 6 | 0 1 0]
[0 9 3 | 0 0 1]

Solve this by reducing the left side (matrix A) into reduced row echelon form. As you do so, the right side (the identity matrix) will transform into A-inverse because you are solving (*).

So the final solution would look like

[1 0 0 | 9/2 -7/4 -1]
[0 1 0 | 1/2 -1/4 0]
[0 0 1 | -3/2 3/4 1/3]

So A-inverse is the right hand side of the augmented matrix.

You can check this by multiplying A and what you get for A-inverse. You should get the identity matrix I.

Or, if you have a graphing calculator (like a TI-89 or similar) you can use the Simult function to plug in A and the identity matrix like we did above and it will do the calculations for you.

These methods will work for square or non-square matrices of any size.

Check the link below for information on reduced row echelon form.

Good luck. Hope that helps.

Answer 2

Hi,

If you have a square matrix A, first find the value of its determinant, which is written as |A|. Then for a 2 by 2 determinant, switch the numbers in the upper left corner and lower right corners with each other. Then change the signs of the other 2 numbers, but leave them in the same positions they already were. Multiply every term of this new matrix by the fraction 1/the determinant |A|. This gives you the new inverse matrix.

There's also a more complicated way to do the inverse of a 3 x 3 matrix at the site below.

Realize that inverses are easy to find on a graphing calculator. If you enter your numbers into matrix A on your calculator, then go back to the main screen and put [A]^-1. This will give you the invers with decimals. [A]^-1>Frac will give the inverse with fractional values. This will work for either a 2 x 2 or 3 x 3 matrix.

I hope that helps.

Answer 3

From the following site you can work out any 2 x 2 and 3 x 3 matrices.

Answer 4

The other answers above are excellent.
Let me show you another method called Faddeev's
method.
Recall that the trace of a matrix A is the sum of its
diagonal elements.
Now let A_1 = A, Tr A = q_1 . B_1 = A_1 -q_1*I

A_2 = AB_1, Tr A_2 = q_2, B_2 = A_2 -q_2*I

------------------------------------------------------------
A_(n -1)= AB_(n-2), TrA_(n-1)/(n-1) = q_(n-1), B_(n-1)=
A_(n-1)- q_(n-1)*I

A_n = AB_(n-1), Tr A_n/n = q_n, B_n = A_n - qn*I

It can be proved that
B_n = 0

and

A^(-1) = B_(n-1)/q_n

if A is nonsingular.
If A is singular, q_n will be 0.
Isn't typing math a scream in this setting?
Sure wish there was a decent way of making
subscripts!
Yuck!!

Answer 5

Inverting a matrix similar to the indentity but just with swapped rows is easy to invert. Whatever rows you would need to swap to make your matrix equal to the identity, you do those swaps on the on the identity, and you'll have your inverse. Say the first matrix is your initial matrix 0001 1000 0100 0100 1000 0010 0010 0001 inverts to 1000 0010 0100 0100 0010 0001 0001 1000 after moving row 1 to row 4, row 3 to row 1 and row 4 to row 3 The 2nd matrix is now your inverse.

Answer 6

You can look for formula for 2 by 2 matrix or do row operations in Gaussian elimination process.

Answer 7

2x2 matrix:

a b
c d
becomes
a -c
-b d
inverse

Answer 8

http://www.easycalculation.com/matrix/learn-matrix-inverse.php

Answer 9

Flip it and reverse it. (Works for 2x2 matrices)

For a larger matrix, split it into smaller 2x2 matrices and then flip and reverse those.

Hope this helps

Answer 10

Yes.