Example 3

Use LU decomposition to find the inverse of

Solution

Knowing that

We can solve for the first column of [B]=by solving for

First solve

, that is

to give

Forward substitution starting from the first equation gives

Hence

Now solve

that is

Backward substitution starting from the third equation gives

=

= -0.9524

= 0.04762

Hence the first column of the inverse of [A] is

Similarly by solving

and solving

gives

Hence

Can you confirm the following for the above example? =

How do I find the inverse of a square matrix using LU Decomposition?

A matrix is the inverse of if . How can we use LU decomposition to find inverse of the matrix? Assume the first column of (the inverse of [A] is

[bb………….b]

then from the above definition of inverse and definition of matrix multiplication.

Similarly the second column of is given by

[A]

Similarly, all columns of can be found by solving n different sets of equations with the column of the right hand sides being the n columns of the identity matrix.