\displaystyle 1.\ \text{If }A=[a_{ij}]\text{ is a square matrix of order }n\text{ and }C_{ij}\text{ denote the cofactor of }a_{ij}\text{ in }A,\text{ then the}
\displaystyle \text{transpose of the matrix of cofactors of elements of }A\text{ is called the adjoint of }A\text{ and is denoted}
\displaystyle \text{by }\mathrm{adj}\ A\ \text{i.e.}\quad \mathrm{adj}\ A=[C_{ij}]^{T}
\displaystyle \text{If }A=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix},\ \text{then }\mathrm{adj}\ A=  \begin{bmatrix} C_{11}&C_{21}&C_{31}\\ C_{12}&C_{22}&C_{32}\\ C_{13}&C_{23}&C_{33} \end{bmatrix}

\displaystyle 2.\ \text{The adjoint of a square matrix of order }2\text{ can be obtained by interchanging the diagonal}
\displaystyle \text{elements and changing the signs of off-diagonal elements.}
\displaystyle \text{i.e. if }A=\begin{bmatrix} a&b\\ c&d \end{bmatrix},\ \text{then }\mathrm{adj}\ A=\begin{bmatrix} d&-b\\ -c&a \end{bmatrix}

\displaystyle 3.\ \text{If }A\text{ is a square matrix of order }n,\text{ then }\ A(\mathrm{adj}\ A)=|A|I_{n}=(\mathrm{adj}\ A)A.

\displaystyle 4.\ \text{Following are some properties of adjoint of a square matrix:}
\displaystyle \text{If }A\text{ and }B\text{ are square matrices of the same order }n,\text{ then}
\displaystyle (i)\ \mathrm{adj}(AB)=(\mathrm{adj}\ B)(\mathrm{adj}\ A)\quad (ii)\ \mathrm{adj}\ A^{T}=(\mathrm{adj}\ A)^{T}\quad
\displaystyle (iii)\ \mathrm{adj}(\mathrm{adj}\ A)=|A|^{\,n-2}A
\displaystyle (iv)\ |\mathrm{adj}\ A|=|A|^{\,n-1}\quad (v)\ |\mathrm{adj}(\mathrm{adj}\ A)|=|A|^{(n-1)^{2}}

\displaystyle 5.\ \text{A square matrix }A\text{ of order }n\text{ is invertible if there exists a square matrix }B
\displaystyle \text{of the same order such that }AB=I_{n}=BA.
\displaystyle \text{In such a case, we say that the inverse of matrix }A\text{ is }B\text{ and we write }A^{-1}=B.
\displaystyle \text{Following are some properties of inverse of a matrix:}
\displaystyle (i)\ \text{Every invertible matrix possesses a unique inverse.}
\displaystyle (ii)\ \text{If }A\text{ is an invertible matrix, then }(A^{-1})^{-1}=A.
\displaystyle (iii)\ \text{A square matrix is invertible iff it is non-singular.}
\displaystyle (iv)\ \text{If }A\text{ is a non-singular matrix, then }A^{-1}=\frac{1}{|A|}(\mathrm{adj}\ A).
\displaystyle (v)\ \text{If }A\text{ and }B\text{ are two invertible matrices of the same order, then }(AB)^{-1}=B^{-1}A^{-1}.
\displaystyle (vi)\ \text{If }A\text{ is an invertible matrix, then }(A^{T})^{-1}=(A^{-1})^{T}.
\displaystyle (vii)\ \text{The inverse of an invertible symmetric matrix is a symmetric matrix.}
\displaystyle (viii)\ \text{If }A\text{ is a non-singular matrix, then }|A^{-1}|=\frac{1}{|A|}.

\displaystyle 6.\ \text{The following are three operations applied on the rows (columns) of a matrix:}
\displaystyle (i)\ \text{Interchange of any two rows (columns).}
\displaystyle (ii)\ \text{Multiplying all elements of a row (column) of a matrix by a non-zero scalar.}
\displaystyle (iii)\ \text{Adding to the elements of a row (column), the corresponding elements of any}
\displaystyle \text{other row (column) multiplied by any scalar.}

\displaystyle 7.\ \text{A matrix obtained from an identity matrix by a single elementary operation}
\displaystyle \text{is called an elementary matrix.}

\displaystyle 8.\ \text{Every elementary row (column) operation on an }m\times n\text{ matrix (not identity matrix)}
\displaystyle \text{can be obtained by pre-multiplication (post-multiplication) with the corresponding}
\displaystyle \text{elementary matrix obtained from the identity matrix }I_{m}\ (I_{n})\text{ by subjecting it}
\displaystyle \text{to the same elementary row (column) operation.}

\displaystyle 9.\ \text{In order to find the inverse of a non-singular square matrix }A\text{ by elementary}
\displaystyle \text{operations, we write }A=IA.
\displaystyle \text{Now we perform a sequence of elementary row operations successively on }A\text{ on the LHS}
\displaystyle \text{and the pre-factor }I\text{ on RHS till we obtain }I=BA.
\displaystyle \text{The matrix }B,\text{ so obtained, is the desired inverse of matrix }A.

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