Inverse Matrix

Describing an Inverse Matrix

• The inverse of a matrix $A$ is a matrix that results in the identity matrix when multiplied by $A$
• Mathematically, the inverse matrix $A^{-1}$ is the matrix that satisfies the following equation:

$$AA^{-1} = I$$

• In other words, an inverse matrix can be thought of the matrix that plays the transformation matrix in reverse
• An inverse matrix is a transformation matrix in itself

Example of an Inverse Matrix

• Let's say our transformation matrix $A$ represents a $90\degree$ clockwise rotation of a vector space:

$$A = \begin{bmatrix} 0 & 1 \cr -1 & 0 \end{bmatrix}$$

• Then, our inverse matrix would look like the following:

$$A^{-1} = \begin{bmatrix} 0 & -1 \cr 1 & 0 \end{bmatrix}$$

• Therefore, $A^{-1}$ represents a $90\degree$ counterclockwise rotation of a vector space

Another Example of an Inverse Matrix

• Let's say our transformation matrix $A$ represents a rightward shear of a vector space:

$$A = \begin{bmatrix} 1 & 1 \cr 0 & 1 \end{bmatrix}$$

• Then, our inverse matrix would look like the following:

$$A^{-1} = \begin{bmatrix} 1 & -1 \cr 0 & 1 \end{bmatrix}$$

• Therefore, $A^{-1}$ represents a leftward shear of a vector space

System of Linear Equations

• A system of linear equations is a collection of one or more linear equations involving the same set of variables
• In linear algebra, we are typically representing a system of linear equations as a linear transformation $Ax = b$

• We can use this system of linear equations $Ax = b$ for two purposes:

  1. Solving for our vector b from the original vector space

     • We typically want to solve for $b$ if we want to see what some vector from a transformed vector space looks like in our original vector space

     • To do this, we need the following:

       • Vector $x$ from the transformed space
       • Transformation matrix $A$ so that we can map vectors from the transformed vector space back to our original vector space
     • Once we have these variables, we can solve for b by just plugging in our $A$ and $x$ into the formula (i.e. multiplying $A$ and $x$ together)

  2. Solving for our initial vector x

     • We typically want to solve for $x$ if we want to see what some vector from our original vector space looks like in a transformed vector space

     • To do this, we need the following:

       • Vector $b$ from our original vector space
       • The inverse of the transformation matrix $A^{-1}$ so that we can map vectors from our original vector space to the transformed vector space
     • Once we have these variables, the equation $Ax = b$ becomes $x = bA^{-1}$ by multiplying each side by $A^{-1}$
     • Then, we can solve for $x$ by just plugging in our $A^{-1}$ and $b$ into the formula (i.e. multiplying $A^{-1}$ and $b$ together)

Solving for the Inverse Matrix

• Suppose we define a square matrix $A$ as the following:

$$A = \begin{bmatrix} a & b \cr c & d \end{bmatrix}$$

• The inverse of a square matrix $A$ can be defined as the following:

$$A^{-1} = \frac{1}{det(A)}\begin{bmatrix} d & -b \cr -c & a \end{bmatrix}$$

• There won't be an inverse of a matrix if the determinant of that matrix is 0

References

• Essence of Linear Algebra Video