The Determinant of a Square Matrix (矩阵的行列式)

The Inverse of a Square Matrix (矩阵的逆)

Definition

For a $n\times n$ square matrix $A$, if there exists a matrix $B$ such that $AB = BA = I_n$, then we call $B$ the inverse of $A$, denoted as $A^{-1}$, and we say that $A$ is invertible.

A matrix is invertible if and only if its determinant is nonzero.

A matrix that is NOT invertible is called a singular matrix, and a matrix that is invertible is called a nonsingular matrix.

Info

For a $2\times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if $det(A) = ad - bc \neq 0$, then $A$ is invertible and

$$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

The inverse of a matrix has the following properties:

1. If $A$ is invertible, then $A^{-1}$ is also invertible and $(A^{-1})^{-1} = A$.
2. For invertible matrices, $(AB)^{-1} = B^{-1}A^{-1}$, $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$, etc.
3. If $A$ is invertible, then $A^T$ is also invertible and $(A^T)^{-1} = (A^{-1})^T$.

Finding the Inverse of a Matrix

An $n \times n$ matrix $A$ is invertible if and only if $A$ is row equivalent to the identity matrix $I_n$. And in this case, we can find $A^{-1}$ by performing the same row operations on $I_n$.

As a result, we can find $A^{-1}$ by row reducing the augmented matrix $\begin{bmatrix} A & I_n \end{bmatrix}$ to the form $\begin{bmatrix} I_n & A^{-1} \end{bmatrix}$.

Copyright

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