❖ In this lesson, we have explored the Matrix inverse and its properties.

Discover what makes the matrix inverse a unique and powerful tool in linear algebra.

Learn through a series of explained properties and see these theories in action through a practical example involving matrix equations.

Matrix inverse and its properties with Examples:

\(1\). \(A^{-1}\) is unique.

\(2\). Reversing an inverse brings you back to the original,

$$(A^{-1})^{-1}=A.$$

\(3\). Scalar impacts on inverse,

$$(rA)^{-1} = (1/r)A^{-1}$$

for non-zero \(r\) in \(\Re.\)

\(4\). Decomposing the inverse of matrix products,

$$(AB)^{-1}=B^{-1}A^{-1},$$

and

$$(ABC)^{-1}=C^{-1}B^{-1}A^{-1}.$$

\(5\). Relationship between transpose and inverse,

$$(A^{-1})^T=(A^T)^{-1}.$$

\(6\). Inverses do not distribute over addition,

$$(A ± B)^{-1} \neq A^{-1} \pm B^{-1}.$$

\(7\). Inverses of powers,

$$A^{-n} = (A^n)^{-1}=(A^{-1})^n.$$

Practical Application:

Find matrix A such that

$$ (I_2 + A)^{-1} =
\left[\begin{array}{cc}
2& 2 \\
2 & 3
\end{array}\right]
$$