ISI2016-DCG-34

Question

Let $A_{ij}$ denote the minors of an $n\times n$ matrix $A.$ What is the relationship  between $\mid A_{ij}\mid$ and $\mid A_{ji}\mid$?

• A. They are always equal.
• B. $\mid A_{ij}\mid=-\mid A_{ji}\mid$ if $i\neq j.$
• C. They are equal if $A$ is a symmetric matrix.
• D. If $\mid A_{ij}\mid=0$ then $\mid A_{ji}\mid=0.$

Comments

Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies.

e.g., $\Delta=\begin{vmatrix} a_{11} &a_{12} &a_{13} \\ a_{21}& a_{22} &a_{23} \\ a_{31} &a_{32} &a_{33} \end{vmatrix}$

minor of $a_{12}:A_{12} = \begin{vmatrix} a_{21} & a_{23} \\ a_{31}&a_{33} \end{vmatrix}$

minor of $a_{21}:A_{21} = \begin{vmatrix} a_{12} & a_{13} \\ a_{32}&a_{33} \end{vmatrix}$  and so on.