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

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 \[\Rightarrow \] \[|adj\,\,A|=0\]