13.1: The Inverse Matrix (aka A^-1)

For some (not all) square matrices \(A\), there exists a special matrix called the Inverse Matrix, which is typically written as \(A^{-1}\) and when multiplied by \(A\) results in the identity matrix \(I\):

\[ A^{-1}A = AA^{-1} = I \nonumber \]

Some properties of an Inverse Matrix include:

1. \((A^{-1})^{-1} = A\)
2. \((cA)^{-1} = \frac{1}{c}A^{-1}\)
3. \((AB)^{-1} = B^{-1}A^{-1}\)
4. \((A^n)^{-1} = (A^{-1})^n\)
5. \((A^\top)^{-1} = (A^{-1})^\top\) here \(A^\top\) is the tranpose of the matrix \(A\).

If you know that \(A^{−1}\) is an inverse matrix to \(A\), then solving \(Ax=b\) is simple, just multiply both sides of the equation by \(A^{−1}\) and you get:

\[A^{-1}Ax = A^{-1}b \nonumber \]

If we apply the definition of the inverse matrix from above we can reduce the equation to:

\[Ix = A^{-1}b \nonumber \]

We know \(I\) times \(x\) is just \(x\) (definition of the identity matrix), so this further reduces to:

\[x = A^{-1}b \nonumber \]

To conclude, solving \(Ax=b\) when you know \(A^{-1}\) is really simple. All you need to do is multiply \(A^{-1}\) by \(b\) and you know \(x\).

Do This

Find a Python numpy command that will calculate the inverse of a matrix and use it invert the following matrix A. Store the inverse in a new matirx named A_inv

import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True) # Trick to make matrixes look nice in jupyter

A = np.matrix([[1, 2, 3], [4, 5, 6], [7,8,7]])

sym.Matrix(A)

#put your answer to the above question here.

Lets check your answer by multiplying A by A_inv.

A * A_inv

np.allclose(A*A_inv, [[1,0,0],[0,1,0],[0,0,1]])

Question

What function did you use to find the inverse of matrix \(A\)?

How do we create an inverse matrix?

From previous assignments, we learned that we could string together a bunch of Elementary Row Operations to get matrix (\(A\)) into it’s Reduced Row Echelon form. We now know that we can represent Elementary Row Operations as a sequence of Elementaary Matrices as follows:

\[ E_n \dots E_3 E_2 E_1 A = RREF \nonumber \]

If \(A\) reduces to the identity matrix (i.e. \(A\) is row equivalent to \(I\)), then \(A\) has an inverse and its inverse is just all of the Elementary Matrices multiplied together:

\[ A^{-1} = E_n \dots E_3 E_2 E_1 \nonumber \]

Consider the following matrix.

\[ A = \left[
\begin{matrix}
1 & 2 \\
4 & 6
\end{matrix}
\right] \nonumber \]

A = np.matrix([[1, 2], [4,6]])

It can be reduced into an identity matrix using the following elementary operators

Words	Elementary Matrix
Adding \(-4\) times row 1 to row 2.	\(E_1 = \left[\begin{matrix}1 & 0 \\ -4 & 1 \end{matrix}\right]\)
Adding row 2 to row 1.	\(E_2 = \left[\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}\right]\)
Multiplying row 2 by \(-\frac{1}{2}\).	\(E_3 = \left[\begin{matrix}1 & 0 \\ 0 & -\frac{1}{2} \end{matrix}\right]\)

E1 = np.matrix([[1,0], [-4,1]])
E2 = np.matrix([[1,1], [0,1]])
E3 = np.matrix([[1,0],[0,-1/2]])

We can just check that the statment seems to be true by multiplying everything out.

E3*E2*E1*A

matrix([[1., 0.],
        [0., 1.]])

Do This

Combine the above elementary Matrices to make an inverse matrix named A_inv

# Put your answer to the above question here.

Do This

Verify that A_inv is an actual inverse and chech that \(AA^{-1} = I\).

# Put your code here.

Question

Is an invertible matrix is always square? Why or why not?

Question

Is a square matrix always invertible? Why or why not?

Question

Describe the Reduced Row Echelon form of a square, invertible matrix.

Question

Is the following matrix in the Reduced Row Echelon form?

\[\begin{split}
\left[
\begin{matrix}
1 & 2 & 0 & 3 & 0 & 4 \\
0 & 0 & 1 & 3 & 0 & 7 \\
0 & 0 & 0 & 0 & 1 & 6 \\
0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right]
\end{split} \nonumber \]

Question

If the matrix shown above is not in Reduced Row Echelon form. Name a rule that is violated?

Question

What is the size of the matrix described in the previous QUESTION?

• \(4 \times 6\)
• \(6 \times 4\)
• \(3 \times 6\)
• \(5 \times 3\)

Question

Describe the elementary row operation that is implemented by the following matrix

\[\begin{split}
\left[
\begin{matrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{matrix}
\right]
\end{split} \nonumber \]