How can I find a matrix's inverse?

Question

Plz go step by step. I am trying to learn but all the websites out there confuse me a bit.
-Math Nerd (aloha91girl)

Answer 1

A matrix is a rectangular array of numbers. For our purposes, let z{m,n} stand for the number z in the mth row and nth column of a matrix.

Example -- Consider the 2x2 matrix named ' A'. Let A be composed of numbers z{1,1}=a, z{1,2}=b, z{2,1}=c, and z{2,2}=d.
Or more simply, the matrix A is set up as
| a b |
| c d |

By definition: If 'A' is square matrix, and if a matrix 'B' can be found such that AB=BA=I, then A is said to be invertible, and B is called an inverse of A.

Now, go back to our Example 2x2 matrix named 'A'. If (ad - bc) is not equal to zero, then the inverse of A, called A^(-1), is equal to

(1/ (ad-bc) ) times the matrix
| d (-b) |
| (-c) a |
where z{1,1}=d, z{1,2}= -b, z{2,1}= -c, and z{2,2}=a.

OR

| (d/(ad-bc)) (-b/(ad-bc)) |
| (-c/(ad-bc)) (a/(ad-bc)) |
where z{1,1} =(d/(ad-bc)), z{1,2} =(-b/(ad-bc)), z{2,1}=(-c/(ad-bc)), and z{2,2} =(a/(ad-bc)).

Answer 2

Using Cramer's rule is one approach, but for larger matrices it is more computationally intensive. A more practical approach from a numerical standpoint is to do Gaussian elimination. Remember when solving a system of linear equations Ax=b, you form the matrix A|b where the column vector b is adjoined as a new column. Performing row operations gets you, assuming the system has a unique solution, to the form I|c, where c is the vector in question. The question of finding a matrix inverse is similar; you are solving an equation AX=I, where X is a matrix and where I is the identity matrix. I'm not going to work any examples, but this should give you a start.

Answer 3

Let me give you an easy way.

| 2 -5 |
| 7 -9 |

To find the inverse, flip and switch. Flip the top left number and the bottom right number:

| -9 -5 |
| 7 2 |

Then switch the sign on the other two numbers:

| -9 5 |
| -7 2 |

The final step is to divide each number by the determinant:

| -9 5 | -18
| -7 2 | -(-35)
=17 <=determinant

| -9/17 5/17 |
| -7/17 2/17 | <=final answer

Answer 4

Let A be the given matrix
1- A must be square matrix
2- Calculate the determinant of A : det(A)
3- If det(a) = 0 then A is a singular matrix and has no inverse
4- If det(a) doesn't =0 so A is a non-singular matrix and have an inverse and we proceed as follows
5- Get At ( the transpose of A by changing rows to columns and vise verse
5- Calculate the adjacent matrix of At by replacing every element in At by its conjugate element ( the value of the determinant you get by eliminating the row and column containing that element multiplied by +ve or -ve sign according to its location ) to get adj(a)
6- A^(-1) = 1/(det(A)) * adj(A)

Answer 5

The condition to find inverse of a matrix
1. it should be a square matrix.
2. it should not be singular, ie det (A) should not be equal to 0.

Now calculate adjoint of the matrix.

A_inverse = 1/det(A) * Adj(A)

Answer 6

This website should help.

This website will tell you how to work inverse matrices
www.analyzemath.com/Calculators/InverseMatrixCalculator.html

also check out www.quickmath.com for a calculator.