Matrix Inverses

1. The Inverse of a Matrix ($A^{-1}$)

In regular math, the inverse of $5$ is $\frac{1}{5}$, because $5 \times \frac{1}{5} = 1$. Matrices have a similar concept!

Definition

Let $A$ be a square matrix. If we can find another matrix $B$ of the same size such that:

Then $A$ is said to be invertible (or non-singular). The matrix $B$ is called the inverse of $A$, and we write it as $A^{-1}$.

Theorem of Uniqueness

If a square matrix $A$ is invertible, it has exactly one unique inverse.

2. Properties of Matrix Inverses

If $A$ and $B$ are invertible matrices of the same size, and $c$ is a non-zero scalar, then:

Double Inverse

The inverse of an inverse is the original matrix.

Scalar Inverse

Watch out! Earlier we learned that a transpose does not flip a scalar: $(cA)^T = cA^T$. But an inverse does flip the scalar!

Product Inverse (The "Socks and Shoes" Rule)

Notice the order flips in reverse! Just like putting on socks then shoes $(AB)$, to reverse the process, you must take off your shoes first, then your socks $(B^{-1} A^{-1})$.

3. Finding the Inverse of a 2x2 Matrix

There is a quick formula to find the inverse of a $2 \times 2$ matrix without doing Gauss-Jordan elimination!

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

First, calculate the value: $ad - bc$. (This is called the determinant).

The Rule

A $2 \times 2$ matrix is invertible if and only if $ad - bc \neq 0$. If it equals zero, stop—the matrix has no inverse!

The Formula

If $ad - bc \neq 0$, you can find the inverse using this formula:

(Swap the items on the main diagonal, and multiply the other two items by -1).

4. Solving Matrix Algebra Equations

When doing algebra with matrices, you must remember two golden rules:

1. You cannot divide by a matrix. Instead, you multiply by its inverse.
2. Order matters. If you multiply the left side of an equation by $A^{-1}$ on the left, you must multiply the right side of the equation by $A^{-1}$ on the left.

Exercise 1: Solve for $X$ in the equation $(X - \mathbb{I}_2)^T = A$

Step 1: Get rid of the transpose by transposing both sides (since $(M^T)^T = M$).

Step 2: Add the identity matrix to both sides.

5. Elementary Matrices and the Invertible Matrix Theorem

Elementary Matrix Definition

An elementary matrix ($E$) is a matrix obtained by performing a SINGLE elementary row operation on the identity matrix.

Elementary Matrix Theorem

Let $E$ be an elementary matrix obtained by performing a specific row operation on $\mathbb{I}_m$. If we take any matrix $A$ and multiply it on the left ($EA$), the resulting matrix is the exact same as if we had performed that row operation directly on $A$.

Theorem: Elementary matrices are always invertible, and their inverses are also elementary matrices!

Row Equivalence

Two matrices $A$ and $B$ are said to be row equivalent if you can transform one into the other by performing a sequence of elementary row operations.

The Invertible Matrix Theorem

Let $A$ be a square matrix. The following statements are mathematically equivalent (if one is true, they are all true!):

1. $A$ is invertible.
2. The equation $AX = \mathbf{0}$ has only the trivial solution (all zeros).
3. The reduced row echelon form (RREF) of $A$ is the identity matrix.
4. $A$ can be expressed as a product of elementary matrices.

6. Algorithm for Finding $A^{-1}$ (Inversion Algorithm)

For matrices larger than $2 \times 2$, we don't have a simple formula. Instead, we use Gauss-Jordan elimination on a super-sized augmented matrix.

The Algorithm

1. Set up an augmented matrix with your matrix $A$ on the left, and the Identity matrix on the right:

2. Perform elementary row operations to reduce the left side ($A$) into the Identity matrix.

3. Whatever operations you perform on the left, you must simultaneously perform on the right side.

4. When the left side becomes $\mathbb{I}$, the right side will have transformed into your inverse matrix, $A^{-1}$: