Q.Solve the following system of equation finding the inverse of coefficient matrix: $x - y = 5$; $2x + 3y = -1$.

What is Matrix Inversion?

Imagine you have a number, say 5. Its multiplicative inverse is $\frac{1}{5}$, because $5 \times \frac{1}{5} = 1$. That "1" is the identity for multiplication — the number that does nothing when you multiply by it.

Matrix inversion is the exact same idea, but for matrices. For a square matrix $A$, its inverse $A^{-1}$ is the matrix that, when multiplied with $A$, gives the identity matrix $I$:

The identity matrix $I$ is the matrix equivalent of the number 1 — it has 1s on the main diagonal and 0s everywhere else. For a $2 \times 2$ matrix, $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

Why Does This Matter?

In algebra, if you have $5x = 20$, you solve for $x$ by multiplying both sides by $\frac{1}{5}$: $x = \frac{1}{5} \times 20 = 4$.

In matrix algebra, if you have $A\mathbf{x} = \mathbf{b}$, you solve for $\mathbf{x}$ by multiplying both sides by $A^{-1}$:

This is how you solve systems of linear equations — the heart of everything from engineering to economics.

The Precise Definition

Let $A$ be an $n \times n$ square matrix. If there exists an $n \times n$ matrix $B$ such that:

then $B$ is called the inverse of $A$, written $A^{-1}$. If such a $B$ exists, $A$ is called invertible or non-singular. If no such $B$ exists, $A$ is singular.

How to Find the Inverse (for $2 \times 2$)

For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

1. Compute the determinant: $\det(A) = ad - bc$
2. If $\det(A) = 0$, stop — no inverse exists.
3. If $\det(A) \neq 0$, the inverse is:

Example

Find the inverse of $A = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$.

$\det(A) = (2)(3) - (1)(5) = 6 - 5 = 1 \neq 0$, so invertible.

Check: $A \times A^{-1} = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix} \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix} = \begin{bmatrix} 6-5 & -2+2 \\ 15-15 & -5+6 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ — it works.

The Big Picture

Matrix inversion is the tool that lets you "divide" by a matrix. It undoes a linear transformation. If $A$ rotates and stretches space, $A^{-1}$ rotates and stretches it back. That's why it only exists when the transformation doesn't collapse anything — when the determinant is non-zero.

The inverse of a matrix $A$ is the unique matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$.