Introduction

When you apply a linear transformation—represented by a matrix—you change vectors: you stretch them, rotate them, shear them, or some combination of these.
But can you undo that transformation?
That is exactly what an inverse matrix does.

This article explains:

• What it means for a matrix to be invertible
• How to compute an inverse
• Why determinants matter
• How inverses relate to “undoing” transformations
• Practical examples and exercises

When Does an Inverse Exist?

A matrix $A$ has an inverse only if:

• $A$ is square (same number of rows and columns)
• $\det(A) \neq 0$

If these conditions hold, we say $A$ is invertible or nonsingular.

If $\det(A) = 0$, the matrix “collapses” space in some way—like flattening 3D space into a plane or flattening a square into a line.
Once information is lost, you cannot undo the transformation.

What the Inverse Matrix Does

If $A$ is a matrix representing a linear transformation, then:

• $A$ sends a vector $x$ to $Ax$
• $A^{-1}$ sends $Ax$ back to $x$

Symbolically: $$A^{-1}(Ax) = x$$ This is why we say $A^{-1}$ “undoes” the action of $A$.

A helpful analogy:

• $A$ is “multiply by 3”
• $A^{-1}$ is “divide by 3”

How to Compute the Inverse of a $2 \times 2$ Matrix

For a matrix $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix},$$ the inverse (when $\det(A) = ad - bc \neq 0$) is: $$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$$ Steps:

• Swap $a$ and $d$
• Negate $b$ and $c$
• Multiply by $\frac{1}{ad - bc}$

This formula is worth memorizing.

How to Compute the Inverse of Larger Matrices

For $3 \times 3$ or bigger matrices, the most common method is row reduction:

1. Write the augmented matrix $(A \mid I)$
2. Perform row operations until the left side becomes $I$
3. The right side becomes $A^{-1}$

Example structure: $$(A \mid I) \longrightarrow (I \mid A^{-1}).$$ This method works for any invertible square matrix.

Geometric Meaning of the Inverse

Every invertible matrix represents a reversible transformation.

Examples:

• A rotation can be undone by rotating the opposite direction.
• A shear can be undone by an “anti-shear.”
• A scaling by factor $k$ can be undone by scaling by $1/k$.

The determinant tells you how the transformation changes area (in 2D) or volume (in 3D).
If the determinant is zero, the transformation squashes space so much that it becomes impossible to reverse.

Examples

Example 1: Inverting a $2 \times 2$ Matrix

Let $$A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}.$$

• Determinant: $2\cdot 2 - 1\cdot 3 = 1$
• Inverse: $$A^{-1} = \begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}.$$

Example 2: Undoing a Transformation

If $A$ sends $$x = \begin{pmatrix} 4 \\ -1 \end{pmatrix}$$ to $$Ax = \begin{pmatrix} 7 \\ 10 \end{pmatrix},$$ then applying $A^{-1}$ recovers $x$: $$A^{-1} \begin{pmatrix} 7 \\ 10 \end{pmatrix} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}.$$

Calculator

Exercises

1. Compute the inverse of $$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}.$$

   Solution

   Matrix: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$
   Determinant: $1\cdot 4 - 2\cdot 3 = -2$
   Inverse: $$\frac{1}{-2} \begin{pmatrix} 4 & -2 \\ -3 & 1 \end{pmatrix}.$$

2. Determine whether the matrix $$\begin{pmatrix} 2 & 6 \\ 1 & 3 \end{pmatrix}$$ is invertible.

   Solution

   Matrix: $\begin{pmatrix} 2 & 6 \\ 1 & 3 \end{pmatrix}$
   Determinant: $2\cdot 3 - 6\cdot 1 = 0$
   Not invertible.

3. Compute the determinant of $$\begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$$ and decide if the inverse exists.

   Solution

   Matrix: $\begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}$
   Determinant: $4\cdot 1 - 1\cdot 2 = 2$
   Nonzero → inverse exists.

4. Use the $2 \times 2$ inverse formula to compute $$A^{-1} \text{ for } A = \begin{pmatrix} 5 & -2 \\ 1 & 1 \end{pmatrix}.$$

   Solution

   Matrix: $\begin{pmatrix} 5 & -2 \\ 1 & 1 \end{pmatrix}$
   Determinant: $5\cdot 1 - (-2)\cdot 1 = 7$
   Inverse: $$\frac{1}{7} \begin{pmatrix} 1 & 2 \\ -1 & 5 \end{pmatrix}.$$

5. True or false: If $\det(A) = 0$, then $A^{-1}$ exists.

   Solution

   False.
   If $\det(A) = 0$, the matrix is not invertible.

6. Suppose $$A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}, \quad x = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.$$ Compute $A^{-1}x$.

   Solution

   $A^{-1} = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}$
   Then: $$A^{-1}x = \begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.$$

7. Using row reduction, find the inverse of $$\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}.$$

   Solution

   Row reduction gives: $$(A \mid I) \to (I \mid A^{-1}) = \left( I \,\middle|\, \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix} \right).$$ So: $$A^{-1} = \begin{pmatrix} 2 & -1 \\ -1 & 1 \end{pmatrix}.$$