Inverses

Inverse of a Matrix
The "opposite" of a matrix or transformation, $A^{-1}$ (inverse of $A$) has the property that applying $A$, then $A^{-1}$ gives the identity transformation. So, $A^{-1}\cdot A=I_n$. Notes:

• $AB=BA=I_n$. $B$ is the inverse of $A$, and $A$ is the inverse of $B$.
• Non-invertible matrices are called singular. Invertible ones are called non-singular.
• Inverses are unique.
• The following properties are true for invertible matrices $A$ and $B$:
  • $(A^{-1})^{-1}=A$
  • $(AB)^{-1}=B^{-1}A^{-1}$
  • $(A^T)^{-1}=(A^{-1})^T$
  • $(\lambda A)^{-1}=\frac{1}{\lambda}A^{-1}$

Computing the Inverse of a Matrix
Lemma 1. A square matrix is invertible if and only if it can be transformed by elementary row operations to row-reduced form with all diagonal entries nonzero.

Lemma 2. Any row-reduced square matrix with all diagonal elements nonzero can be transformed to the identity matrix with elementary row operations.

By these lemmas, we can write $$E_{k}E_{k-1}\dots E_2E_1A=I_n,$$ so $A^{-1}=E_{k}E_{k-1}\dots E_2E_1$. So, we can compute the inverse of matrix $A$ by applying the elementary row operations $E_k,\dots, E_1$ to the identity matrix! So, to compute $A^{-1}$, apply the same row operations to the identity matrix as you applied to $A$ in order to transform it to the identity. One way of finding the inverse of an $n\times n$ matrix $A$ is as follows:

1. Form the $n\times 2n$ matrix $[A\,|\,I_n]$.
2. Row-reduce the matrix to $[I_n\,|\,A^{-1}]$.
3. The matrix $A^{-1}$ is the $n\times n$ matrix on the right side of the augmented matrix.

Ex. $$\begin{align*}\text{Find the inverse of }&\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}.\\\left[\begin{array}{cc|cc}1 & 2 & 1 & 0\\3 & 4 & 0 & 1\end{array}\right]&\rightarrow\left[\begin{array}{cc|cc}1 & 2 & 1 & 0\\0 & -2 & -3 & 1\end{array}\right]\rightarrow\\\left[\begin{array}{cc|cc}1 & 2 & 1 & 0\\0 & 1 & 3/2 & -1/2\end{array}\right]&\rightarrow\left[\begin{array}{cc|cc}1 & 0 & -2 & 1\\0 & 1 & 3/2 & -1/2\end{array}\right]\rightarrow\\\text{So, the inverse is }&\boxed{\begin{bmatrix}-2 & 1\\3/2 & -1/2\end{bmatrix}}.\end{align*}$$ It's good practice to verify results, so you should multiply the inverse by the original matrix to check that the result is the identity matrix.

As a side note, for $2\times 2$ matrices, we can find the determinant directly with the formula: $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d & -b\\ -c & a\end{bmatrix}.$$

Invertible Matrix Theorem
For any $n\times n$ matrix $A$, the following are equivalent:

1. $A$ is invertible.
2. $A$ is row equivalent to $I_n$.
3. $A$ has $n$ pivot positions.
4. The equation $Ax=0$ has only the trivial solution.
5. The columns of $A$ are linearly independent.
6. The equation $Ax=b$ has at least one solution for each $b$ in $\mathbb{R}^n$.
7. The columns of $A$ span $\mathbb{R}^n$.
8. The equation $Ax=b$ has exactly one solution for each $b$ in $\mathbb{R}^n$.
9. The determinant of $A$ is nonzero.
10. The transpose of $A$ is invertible.

You don't need to memorize each one of these, but do take the time to understand why they are true — this will help you remember them.