Lab 11: Inverting matrices

In this lab you will use Sympy Matrix objects to write a program that will invert a matrix. Note that Sympy Matrix objects behave a little differently than Numpy matrix objects. You can import Sympy using the following command:

>>> import sympy as sp

Rows and entries of Sympy Matrix objects can be accessed in the following ways.

>>> M=sp.Matrix([[1,2,3],[4,5,6]])
>>> M.row(1)
Matrix([[4, 5, 6]])
>>> M[1,0]
4
>>> M[4]
5

Notice that if you only use a single bracket index you get the entry as though the matrix were a single list, not the row! The dimensions of the matrix can be accessed using the shape property in the same way you would for a Numpy matrix object. Remember, this is not a method so do not use parentheses.

>>> M.shape
(2,3)

You can learn more about the Sympy Matrix object by reading https://docs.sympy.org/0.7.2/modules/matrices/matrices.html or the Acme introduction to Sympy in Chapter 10 of Python Essentials: https://foundations-of-applied-mathematics.github.io/.

Task 1

Write a function aug_id(A) that takes as input an m x n matrix object A and augments the matrix by the identity matrix. There is no built-in Sympy function to do this, so you will need to create a new matrix object of the appropriate size and edit its entries.

>>> A=sp.Matrix([[1,2],[3,4]])
>>> aug_id(A)
Matrix([
[ 1,  2, 1, 0],
[ 3,  4, 0, 1]])
>>> B=sp.Matrix([[0,2],[3,4],[5,6]])
>>> aug_id(B)
Matrix([
[ 0,  2, 1, 0, 0],
[ 3,  4, 0, 1, 0],
[ 5,  6, 0, 0, 1]])

Task 2

Write a function mat_inv(A) that takes as input an m x n matrix object A and returns a potential inverse. Use the function from part one to augment the input matrix by the identity, then use the rref method (see below) to row reduce the augmented matrix. Your function should then extract the matrix consisting of the final n columns of the row-reduced matrix. If A is invertible, this will be the inverse. Your function should still return a matrix even if the input matrix was not invertible.

Sympy Matrix objects have a built-in method rref that returns the matrix in reduced echelon form.

>>> A=sp.Matrix([[1,2],[3,4]])
>>> aug_id(A).rref()
(Matrix([
 [1, 0,  -2,    1],
 [0, 1, 3/2, -1/2]]),
 (0, 1))

The first returned value is the reduced matrix; the second is a tuple containing the indices of the columns containing leading terms.

>>> A=sp.Matrix([[1,2],[3,4]])
>>> mat_inv(A)
Matrix([
[ -2,    1],
[3/2, -1/2]])
>>> B=sp.Matrix([[1,2,3],[4,5,6]])
>>> mat_inv(B)
Matrix([
[-5/3,  2/3],
[4/3,  -1/3]])

Challenge

Write a function inv_true(A) which calculates the potential inverse as in part 2, and then verifies that the potential inverse is a true inverse for the input matrix.