21-06-2024 06:40

Status :

Tags : Matrix linear algebra

Inverse of a matrix

If $A$ and $B$ are two non-singular square matrices of the same order such that $AB=BA=\mathbf{I}$,

where $\mathbf{I}$ is theidentity matrix, then $B$ is called the inverse of $A$, i.e. $B=A^{-1}$. We say $A^{-1}$ is the inverse of the matrix $A$ if $A{A}^{-1}=A^{-1}A=\mathbf{I}$.

To find the inverse matrix with the help of Adjoint matrix

$A^{-1}=\frac{\text{adj}(A)}{|A|}$

Note:

• The necessary condition for a square matrix $A$ to possess an inverse is that the matrix $A$ is non-singular i.e., $|A|\ne 0$.
• A square non-singular matrix $A$ of order $n$ is said to be invertible if there exists a non-singular square matrix $B$ of order $

Properties of Inverse Matrices

• If $A^{-1}$ exists, then it is unique. • ${\left(A^{-1}\right)}^{-1}=A$. • ${\left(A^T\right)}^{-1}={\left(A^{-1}\right)}^T$. • Let $D=\text{diag}(d_{11},d_{22},d_{33},d_{44},\dots ,d_{nn})$, $d_{ii}\ne 0$. Then $D^{-1}=\text{diag}\left(\frac{1}{d_{11}},\frac{1}{d_{22}},\frac{1}{d_{33}},\dots ,\frac{1}{d_{nn}}\right)$. • The inverse of a non-singular symmetric matrix is a symmetric matrix. • ${\left(A^{-1}\right)}^n=A^{-n}$.

References