ISI2015-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. $\mid A_{ij} \mid = – \mid A _{ji} \mid \text{ if } i \neq j$
• C. They are equal if $A$ is a symmetric matrix
• D. If $\mid A_{ij} \mid =0$ then $\mid A_{ji} \mid =0$

Answer 1

Let $A = {\begin{pmatrix} a &b \\ c & d \end{pmatrix}} $

$A_{11} = d$,$A_{12} = c$,$A_{21} = b$,$A_{22} = a$

$\implies |A_{11}| = d$,$|A_{12}| = c$,$|A_{21}| = b$,$|A_{22}| = a$

 

As we can see $|A_{12}| = |A_{21}|$ if $c=b$

$\implies$ if matrix is ${\begin{pmatrix} a &c \\ c & d \end{pmatrix}} $

$\implies A$ should be a symmetric matrix.

 

$\therefore$ Option $C.$ is correct answer

Comments

correct