JEE Main & Advanced Mathematics Determinants & Matrices Inverse of a Matrix

A non-singular square matrix of order \[n\] is invertible if there exists a square matrix B of the same order such that \[AB={{I}_{n}}=BA\].

In such a case, we say that the inverse of A is B and we write \[{{A}^{-1}}=B\]. The inverse of A is given by \[{{A}^{-1}}=\frac{1}{|A|}.adj\,A\].

The necessary and sufficient condition for the existence of the inverse of a square matrix A is that \[|A|\ne 0\].

Properties of inverse matrix:

If A and B are invertible matrices of the same order, then 

(i) \[{{({{A}^{-1}})}^{-1}}=A\]

(ii) \[{{({{A}^{T}})}^{-1}}={{({{A}^{-1}})}^{T}}\]

(iii) \[{{(AB)}^{-1}}={{B}^{-1}}{{A}^{-1}}\]               

(iv) \[{{({{A}^{k}})}^{-1}}={{({{A}^{-1}})}^{k}},k\in N\]   [In particular \[{{({{A}^{2}})}^{-1}}={{({{A}^{-1}})}^{2}}]\]

(v) \[adj({{A}^{-1}})={{(adj\,A)}^{-1}}\]

(vi) \[|{{A}^{-1}}|\,=\frac{1}{|A|}=\,|A{{|}^{-1}}\]

(vii) A = diag \[({{a}_{1}}{{a}_{2}}...{{a}_{n}})\]\[\Rightarrow {{A}^{-1}}=diag\,(a_{1}^{-1}a_{2}^{-1}...a_{n}^{-1})\]

(viii)  A is symmetric \[\Rightarrow \] \[{{A}^{-1}}\] is also symmetric.

(ix) A is diagonal, \[|A|\ne 0\,\,\Rightarrow {{A}^{-1}}\]is also diagonal.

(x) A is a scalar matrix \[\Rightarrow \] \[{{A}^{-1}}\]is also a scalar matrix.

(xi) A is triangular, \[|A|\ne 0\]\[\rightleftharpoons \]\[{{A}^{-1}}\]is also triangular.    

(xii) Every invertible matrix possesses a unique inverse.

(xiii)  Cancellation law with respect to multiplication

If A is a non-singular matrix i.e., if \[|A|\ne 0\], then \[{{A}^{-1}}\]exists and \[AB=AC\Rightarrow {{A}^{-1}}(AB)={{A}^{-1}}(AC)\]

\[\Rightarrow \] \[({{A}^{-1}}A)B=({{A}^{-1}}A)C\]

\[\Rightarrow \] \[IB=IC\Rightarrow B=C\]

\[\therefore \] \[AB=AC\Rightarrow B=C\Leftrightarrow |A|\,\ne 0\].