Invertible Matrix

Table of contents
  1. Inverse Matrix
    1. Matrix of a System of Linear Equations
      1. Homogeneous System of Linear Equations
    2. Singular matrix is not invertible
    3. Properties of inverse matrix
  2. How to find the inverse matrix
    1. Using determinant and adjugate matrix
    2. Using elementary row operations
    3. Gaussian Elimination on Augmented Matrix
    4. Using LU decomposition

Inverse Matrix

Inverse matrix of a $n \times n$ square matrix $A$ is denoted by $A^{-1}$ and defined as the matrix that satisfies the following equation:

$$ AA^{-1} = A^{-1}A = I_n $$

If a matrix has an inverse matrix, it is called invertible, regular, or non-singular.

Matrix of a System of Linear Equations

If a matrix $A$ is invertible, then the system of linear equations $Ax = b$ has a unique solution $x = A^{-1}b$.

If a matrix is singular, then the system of linear equations $Ax = b$ has either no solution or infinitely many solutions.

Homogeneous System of Linear Equations

For a homogeneous system of linear equations $Ax = 0$, if $A$ is invertible, then the only solution is $x = 0$.

$x = 0$ in a homogeneous system of linear equations is called the trivial solution.

Singular matrix is not invertible

A matrix is singular if and only if its determinant is zero.

If the determinant is zero, then the matrix is not invertible.

See down below. You’ll see that the inverse matrix is undefined when the determinant is zero.

Properties of inverse matrix

For any invertible matrix $A$ and $B$ of the same size and scalar $c \neq 0$,

  • $(A^{-1})^{-1} = A$
  • $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$
  • $(cA)^{-1} = \frac{1}{c}A^{-1}$
  • $A^k$ is invertible and $(A^k)^{-1} = (A^{-1})^k$ for $k > 0 \wedge k \in \mathbb{Z}$
Invertible Matrix Theorem

For a square matrix $A \in \mathbb{R}^{n \times n}$, the following statements are all equivalent:

  • $A$ is invertible.
  • $A$ is non-singular.
  • $A$ is row equivalent to $I_n$.
  • $A$ in a row echelon form has $n$ pivots.
  • The system of linear equations $Ax = b$ has a unique solution for every $b \in \mathbb{R}^n$.
  • The homogeneous system of linear equations $Ax = 0$ has only the trivial solution.
  • The columns of $A$ are linearly independent $\Leftrightarrow \rank(A) = n$
  • The columns of $A$ span $\mathbb{R}^n$ $\Leftrightarrow \dim(\text{Im}(A)) = n$
  • $A^T$ is invertible.

How to find the inverse matrix

Using determinant and adjugate matrix

If a square matrix $A$ is invertible, then the inverse matrix $A^{-1}$ can be found by

$$ A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A) $$

where $\det(A)$ is the determinant of $A$ and $\operatorname{adj}(A)$ is the adjugate matrix of $A$.

Using elementary row operations

If we can transform a matrix $A$ into an identity matrix $I_n$ using a sequence of elementary row operations, then the matrix multiplication of the corresponding elementary matrices is the inverse matrix $A^{-1}$:

$$ E_k \cdots E_2 E_1 A = I_n \Rightarrow A^{-1} = E_k \cdots E_2 E_1 $$

Gaussian Elimination on Augmented Matrix

Pretty much the same idea as above.

We want to find matrix $X$ s.t. $AX = I$.

So we augment $A$ with $I_n$ and perform Gaussian elimination on it:

\[[A \mid I_n] \xrightarrow{\text{Gaussian elimination}} [I_n \mid X]\]

Then $X = A^{-1}$.

Using LU decomposition

To be added