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

Inverse of a Matrix

Category : JEE Main & Advanced

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 $AB=AC\Rightarrow {{A}^{-1}}(AB)={{A}^{-1}}(AC)$

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

$\Rightarrow$ $IB=IC\Rightarrow B=C$

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

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