Inverse Matrices

The Idea of Reversibility

Imagine you apply a transformation to space—stretching, rotating, shearing. Now ask yourself: can you undo it? Can you find another transformation that takes you back exactly to where you started?

This is the central question behind inverse matrices. If a matrix $A$ represents some transformation, then its inverse $A^{-1}$ is the transformation that perfectly reverses it. Apply $A$ and then $A^{-1}$, and you end up exactly where you began.

Think of it like this: if $A$ is a recipe that transforms ingredients into a dish, then $A^{-1}$ is the reverse recipe that extracts the original ingredients from the dish. Of course, not every cooking process can be reversed—some transformations destroy information. The same is true for matrices.

Forward and Reverse

Let's see this in action. Below, we start with the standard basis vectors $\vec{i}$ and $\vec{j}$, apply a transformation $A$, and then apply its inverse $A^{-1}$ to get back to the original.

Notice how applying $A^{-1}$ after $A$ brings us right back to the identity—the original unit vectors. This is the defining property of an inverse:

Here, $I$ is the identity matrix, which represents doing nothing to space. When you multiply any vector by $I$, it stays exactly where it is.

Solving Systems of Equations

Why do we care about inverses? One of the most powerful applications is solving systems of linear equations. Consider the equation:

This says: some unknown vector $\vec{x}$, when transformed by $A$, lands on $\vec{b}$. To find $\vec{x}$, we need to reverse the transformation. Multiply both sides by $A^{-1}$:

The inverse tells us exactly where $\vec{x}$ was before the transformation—the answer to our equation.

Drag the sliders to change $\vec{b}$ and watch how the solution $\vec{x}$ changes. The blue vector shows where $\vec{x}$ must be so that after transformation by $A$, it lands on the purple $\vec{b}$.

When Does an Inverse Exist?

Here's the crucial question: when can a transformation be reversed? The answer connects directly to the determinant.

Remember that the determinant measures how a transformation scales area. If the determinant is zero, the transformation squashes space down to a lower dimension—a line, or even a single point. When this happens, information is lost. Multiple input vectors get mapped to the same output, so there's no way to uniquely trace back.

When $k = 2$, the two columns of the matrix become parallel—they point in the same direction. The entire 2D plane gets squashed onto a single line. This matrix is called singular, and it has no inverse.

The rule is simple:

A non-zero determinant means the transformation preserves dimensionality—nothing is lost, so everything can be recovered.

The Formula for 2×2 Inverse

For a 2×2 matrix, there's a beautiful explicit formula. Given:

The inverse is:

Notice the pattern: swap the diagonal entries ($a$ and $d$), negate the off-diagonal entries ($b$ and $c$), and divide everything by the determinant $ad - bc$.

This formula makes it clear why the determinant matters: we're dividing by it. If $ad - bc = 0$, the formula breaks down—you can't divide by zero.

Play with the matrix entries. Watch how the inverse changes, and verify that $A \cdot A^{-1} = I$. Try making the determinant approach zero and see what happens.

Geometric Intuition

Here's another way to think about the inverse. If $A$ stretches space by a factor of 2 along some direction, then $A^{-1}$ compresses it by a factor of 1/2 along that same direction. If $A$ rotates by 30° counterclockwise, $A^{-1}$ rotates by 30° clockwise.

Every action has an equal and opposite reaction. The inverse is the "anti-transformation" that undoes exactly what the original did.

This also explains why singular matrices have no inverse: if $A$ flattens a 2D region to a 1D line, there's no way to "unflatten" it back. The information about where points were in the perpendicular direction is gone forever.

Key Takeaways

• The inverse $A^{-1}$ is the transformation that perfectly reverses $A$
• The defining property: $A \cdot A^{-1} = I$, the identity matrix
• An inverse exists if and only if $\det(A) \neq 0$
• When $\det(A) = 0$, the matrix is singular—it squashes space and loses information
• We can solve $A\vec{x} = \vec{b}$ by computing $\vec{x} = A^{-1}\vec{b}$
• For 2×2 matrices: swap diagonals, negate off-diagonals, divide by the determinant