Inverse+of+matrices

__The Inverse of a Matrix__
__**DEFINITION:**__ Assuming we have a square matrix A, which is non-singular (i.e. det(A) does not equal zero), then there exists an n×n matrix A-1 which is called the inverse of A, such that this property holds: AA-1 = A-1A = I, where I is the identity matrix.

__The inverse of a 2×2 matrix__
Take for example a arbitury 2×2 Matrix A whose determinant (ad − bc) is not equal to zero. where a,b,c,d are numbers, The inverse is: Now try finding the inverse of your own 2×2 matrices.

__The inverse of a n×n matrix__
The inverse of a general n×n matrix A can be found by using the following equation. Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the following method: Lastly to find the inverse of A divide the matrix CT by the determinant of A to give its inverse. Now test this method with finding the inverse of your own 3×3 matrice.
 * Given the n×n matrix A, defineB = //bij//to be the matrix whose coefficients are found by taking the determinant of the (n-1) × (n-1) matrix obtained by deleting the ith row and jth column of A. The terms of B (i.e. B = //b//ij) are known as the cofactors of A.
 * Define the matrix C, where //cij// = (−1)//i+j// //bij//.
 * The transpose of C (i.e. CT) is called the adjoint of matrix A.

__Solving Systems of Equations using Matrices__
__**DEFINITION:**__ A system of linear equations is a set of equations with n equations and n unknowns, is of the form of The unknowns are denoted by x1, x2, ..., xn and the coefficients (a and b above) are assumed to be given. In matrix form the system of equations above can be written as: A simplified way of writing above is like this: A x = b Now, try putting your own equations into matrix form.

After looking at this we will now look at two methods used to solve matrices. These are:
 * Inverse Matrix Method
 * Cramer's Rule

__Inverse Matrix Method__
or alternatively So by calculating the inverse of the matrix and multiplying this by the vector b we can find the solution to the system of equations directly. And from earlier we found that the inverse is given by From the above it is clear that the existence of a solution depends on the value of the determinant of A. There are three cases: Looking at two equations we might have that Written in matrix form would look like and by rearranging we would get that the solution would look like Now try solving your own two equations with two unknowns. Similarly for three simultaneous equations we would have: Written in matrix form would look like and by rearranging we would get that the solution would look like Now try solving your own three equations with three unknowns.
 * DEFINITION:** The inverse matrix method uses the inverse of a matrix to help solve a system of equations, such like the above A x = b . By pre-multiplying both sides of this equation by A-1 gives:
 * 1) If the det(A) does not equal zero then solutions exist using [[image:http://www.maths.surrey.ac.uk/explore/emmaspages/images/matrices/Mat26.gif caption="Ax=b derivation"]]
 * 2) If the det(A) is zero and b=0 then the solution will be not be unique or does not exist.
 * 3) If the det(A) is zero and b=0 then the solution can be x = 0 but as with 2. is not unique or does not exist.

__Cramer's Rule__
The first term x1 above can be found by replacing the first column of A by Doing this we obtain: Similarly for the general case for solving xr we replace the rth column of A by expand the determinant. This method of using determinants can be applied to solve systems of linear equations. We will illustrate this for solving two simultaneous equations in x and y and three equations with 3 unknowns x, y and z.
 * DEFINITION:** Cramer's rule uses a method of determinants to solve systems of equations. Starting with equation below,

Two simultaneous equations in x and y
To solve use the following: or simplified: Now try solving two of your own equations.

Three simultaneous equations in x, y and z
ax + by + cz = p dx + ey + fz = q gx + hy + iz = r To solve use the following: