# JEE Main & Advanced Mathematics Matrices Adjoint of a Square Matrix

## Adjoint of a Square Matrix

Category : JEE Main & Advanced

Let $A=[{{a}_{ij}}]$be a square matrix of order $n$ and let ${{C}_{ij}}$be cofactor of ${{a}_{ij}}$in  A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A

Thus, $adj$$A={{[{{C}_{ij}}]}^{T}}\Rightarrow {{(adj\,A)}_{ij}}={{C}_{ji}}=$cofactor of ${{a}_{ji}}$in A.

If $A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}} \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}} \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}} \\\end{matrix} \right],$ then  $adj\,A={{\left[ \begin{matrix} {{C}_{11}} & {{C}_{12}} & {{C}_{13}} \\ {{C}_{21}} & {{C}_{22}} & {{C}_{23}} \\ {{C}_{31}} & {{C}_{32}} & {{C}_{33}} \\\end{matrix} \right]}^{T}}=\left[ \begin{matrix} {{C}_{11}} & {{C}_{21}} & {{C}_{31}} \\ {{C}_{12}} & {{C}_{22}} & {{C}_{32}} \\ {{C}_{13}} & {{C}_{23}} & {{C}_{33}} \\\end{matrix} \right];$ where ${{C}_{ij}}$denotes the cofactor of ${{a}_{ij}}$in A.

Example : $A=\left[ \begin{matrix} p & q \\r & s \\\end{matrix} \right],\,{{C}_{11}}=s,\,{{C}_{12}}=-r,\,{{C}_{21}}=-q,\,{{C}_{22}}=p$ $\therefore adj\,A={{\left[ \begin{matrix} s & -r \\ -q & p \\\end{matrix} \right]}^{T}}=\left[ \begin{matrix} s & -q \\ -r & p \\\end{matrix} \right]$

Properties of adjoint matrix : If A, B are square matrices of order $n$ and ${{I}_{n}}$is corresponding unit matrix, then

(i) $A(adj\,A)=|A|{{I}_{n}}=(adj\,A)A$

(Thus A (adj A) is always a scalar matrix)

(ii) $|adj\,A|=|A{{|}^{n-1}}$

(iii) $adj\,(adj\,A)=|A{{|}^{n-2}}A$

(iv) $|adj\,(adj\,A)|\,=\,|A{{|}^{{{(n-1)}^{2}}}}$

(v) $adj\,({{A}^{T}})={{(adj\,A)}^{T}}$

(vi) $adj\,(AB)=(adj\,B)(adj\,A)$

(vii) $adj({{A}^{m}})={{(adj\,A)}^{m}},m\in N$

(viii) $adj(kA)={{k}^{n-1}}(adj\,A),k\in R$

(ix) $adj\,({{I}_{n}})={{I}_{n}}$

(x) $adj\,(O)=O$

(xi) A is symmetric $\Rightarrow$ adj A is also symmetric.

(xii) A is diagonal $\Rightarrow$ adj A is also diagonal.

(xiii) A is triangular $\Rightarrow$ adj A is also triangular.

(xiv) A is singular $\Rightarrow$ $|adj\,\,A|=0$

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