13.1: The Inverse Matrix (aka A^-1)
- Page ID
- 65482
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:
- \((A^{-1})^{-1} = A\)
- \((cA)^{-1} = \frac{1}{c}A^{-1}\)
- \((AB)^{-1} = B^{-1}A^{-1}\)
- \((A^n)^{-1} = (A^{-1})^n\)
- \((A^\top)^{-1} = (A^{-1})^\top\) here \(A^\top\) is the 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 \(x\).
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
Lets check your answer by multiplying A
by A_inv
.
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.
\[ A = \left[
\begin{matrix}
1 & 2 \\
4 & 6
\end{matrix}
\right] \nonumber \]
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 \(-\frac{1}{2}\). | \(E_3 = \left[\begin{matrix}1 & 0 \\ 0 & -\frac{1}{2} \end{matrix}\right]\) |
We can just check that the statment seems to be true by multiplying everything out.
Combine the above elementary Matrices to make an inverse matrix named A_inv
Verify that A_inv
is an actual inverse and chech that \(AA^{-1} = I\).
Is an invertible matrix is always square? Why or why not?
Is a square matrix always invertible? Why or why not?
Describe the Reduced Row Echelon form of a square, invertible matrix.
Is the following matrix in the Reduced Row Echelon form?
\[\begin{split}
\left[
\begin{matrix}
1 & 2 & 0 & 3 & 0 & 4 \\
0 & 0 & 1 & 3 & 0 & 7 \\
0 & 0 & 0 & 0 & 1 & 6 \\
0 & 0 & 0 & 0 & 0 & 0
\end{matrix}
\right]
\end{split} \nonumber \]
If the matrix shown above is not in Reduced Row Echelon form. Name a rule that is violated?
What is the size of the matrix described in the previous QUESTION?
- \(4 \times 6\)
- \(6 \times 4\)
- \(3 \times 6\)
- \(5 \times 3\)
Describe the elementary row operation that is implemented by the following matrix
\[\begin{split}
\left[
\begin{matrix}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 1
\end{matrix}
\right]
\end{split} \nonumber \]