6.1 Properties of Inverse of a Matrix

  Matrices and Determinants

• If $A$ & $B$ are invertible matrices of the same order,then $AB$ is invertible and ${\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}$.
• If $A$ is an invertible square matrix ,then ${A^T}$ is also invertible and ${\left( {{A^T}} \right)^{ - 1}} = {\left( {{A^{ - 1}}} \right)^T}$.
• If $A$ is a non-singular square matrix of order $n$ ,then $\left| {adjA} \right| = {\left| A \right|^{n - 1}}$
• If $A$ is an invertible square matrix ,then adj$${adj\left( {{A^T}} \right) = {{\left( {adjA} \right)}^T}}$$.
• $A$ and $B$ are non-singular square matrices of the same order,then ${adj\left( {AB} \right) = \left( {adjA} \right)(adjB)}$
• If $A$ is a non-singular square matrix ,then $adj\left( {adjA} \right) = {\left| A \right|^{n - 2}}A$.

Example:

For two non singular matrices $AB$ ,show that $adj\left( {AB} \right) = \left( {adjB} \right)\left( {adjA} \right)$

Solution:

We have $\left( {AB} \right)\left( {adj\left( {AB} \right)} \right) = \left| {AB} \right|{I_n}$

$ = \left| A \right|\left| B \right|{I_n}$

${A^{ - 1}}\left( {AB} \right)\left( {adj\left( {AB} \right)} \right) = \left| A \right|\left| B \right|{A^{ - 1}}$

$ \Rightarrow B.adj\left( {AB} \right) = \left| B \right|.adjA\left( {\because {A^{ - 1}} = \frac{1}{{\left| A \right|}}adjA} \right)$

$ \Rightarrow {B^{ - 1}}B.adj\left( {AB} \right) = \left| B \right|{B^{ - 1}}.adjA$

$ \Rightarrow adj\left( {AB} \right) = \left( {adjB} \right)\left( {adjA} \right)$