07 Pre-Class Assignment: Inverse and Transformation Matrices#

Readings for this topic (Recommended in bold)#

Assignment Overview#

Transpose
Inverse
Transformations

1. The Transpose of a Matrix#

Definition Given an $m\times n$ matrix $A$, the transpose of $A$ is the $(n\times m)$ matrix $A^T$, defined by $$ [A^T]_{ij} = [A]_{ji}, $$ for $1\leq i\leq m, 1\leq j\leq n$. For example

\[\begin{split} A = \left[ \begin{matrix} 1 & 2\\ 3 & 4 \end{matrix} \right] \qquad\Longrightarrow \qquad A^T = \left[ \begin{matrix} 1 & 3\\ 2 & 4 \end{matrix} \right] \end{split}\]

\[\begin{split} B = \left[ \begin{matrix} 1 & 2 & 30\\ 3 & 4 & 9 \end{matrix} \right] \qquad\Longrightarrow \qquad B^T = \left[ \begin{matrix} 1 & 3\\ 2 & 4 \\ 30 & 9 \end{matrix} \right] \end{split}\]

Note that the rows of $A^T$ are the columns of $A$, and the columns of $A^T$ are the rows of $A$, i.e., the transpose operation swaps rows and columns.

In Python, we can find the transpose of a matrix A (numpy array or matrix) by using A.T

✅ QUESTION: Create the following matrix in numpy, and then find its transpose $$ \left[ \begin{matrix} 1 & 0 & 0\\ 2 & 3 & 0\\ 1 & 3 & 4\\ 1 & 9 & 7\\ \end{matrix} \right] $$

Store the transposed matrix in a variable called Atranspose for the answercheck.

# Put your code here

from answercheck import checkanswer
checkanswer.matrix(Atranspose,"c280f7e2eaf174101b162d78350dce37")

2. The Inverse Matrix (a.k.a. $A^{-1}$)#

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 $I$:

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

Note that this is defined only for square matrices and thus, the matrices $A$, $A^{-1}$, and $I$ above all have the same size.

Some properties of an Inverse Matrix include:

$(A^{-1})^{-1} = A$
$(cA)^{-1} = \frac{1}{c}A^{-1}$
$(AB)^{-1} = B^{-1}A^{-1}$
$(A^n)^{-1} = (A^{-1})^n$
$(A^\top)^{-1} = (A^{-1})^\top$ here $A^\top$ is the transpose of the matrix $A$.

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

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

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

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

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

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

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.

Let’s 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 $A$?

Put your answer to the above question here

✅ DO THIS: Now, consider the following matrix B. Try to find its inverse. Explain why you are getting the result you are getting.

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

B = np.matrix([[1, 2, 3], [4, 5, 6], [2,4,6]])

sym.Matrix(B)

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 its Reduced Row Echelon Form. We later discussed that each Elementary Row Operation can be represented by a multiplication by an Elementary Matrix. Thus, the Gauss-Jordan Elimination (using Elementary Row Operations) can be represented as multiplication by a number of Elementary Matrices as follows:

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

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

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

Consider the following matrix.
$$ A = \left[ \begin{matrix} 1 & 2 \\ 4 & 6 \end{matrix} \right] $$

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

✅ 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 by checking that $A^{-1}A = I$ and $AA^{-1} = I$.

# Put your code here.

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