Linear Algebra 2.2 - 2.3 Inverse of Matrix

Section 2.2 Inverse of Matrix

An $n\times n$ matrix $A$ is said to be invertible if there is an $n\times n$ matrix $C$ such that :

CA=I\and AC=I

If we change \and to \or, it is also right. But it is not the definition.

In this case is case, $C$ is an inverse of $A$ , and is uniquely determined by $A$ , we can write in $A^{-1}$ .

Theorem

The quantity $ad-bc$ is called the determinant of $A$ , we can write :

$\det A=ad-bc$

Theorem

If $A$ is an invertible $n\times n$ matrix, then for each $b$ in $\R^n$ , the equation $Ax=b$ has the unique solution $x=A^{-1}b$ .

Theorem

If $A$ is an invertible matrix, then $A^{-1}$ is invertible and $(A^{-1})^{-1}=A$
If $A$ and $B$ are $n\times n$ invertible matrices, then so is $AB$ . Then $(AB)^{-1}=B^{-1}A^{-1}$
If $A$ is an invertible matrix, then $(A^T)^{-1}=(A^{-1})^T$

An invertible matrix $A$ is row equivalent to a identity matrix.

Elementary Matrices

An elementary matrix is one that is obtained by performing a single elementary row operation on an identity matrix.

If an elementary row operation is performed on an $m\times n$ matrix $A$ , the resulting matrix can be written as $EA$ , where the $m\times m$ matrix $E$ is created by performing the same row operation on $I_m$ .

Each elementary matrix $E$ is invertible. The inverse of $E$ is the elementary matrix of the same type that transforms $E$ back to $A$ .

Theorem

An $n\times n$ matrix $A$ is invertible, then any sequence of elementary row operations that reduces $A$ to $I_n$ also transforms $I_n$ to $A^{-1}$ .