Question

Let $A=\left[a_{ij}\right]$ be a matrix of order $3\times 3$.
Assertion: Expansion of determinant of A along second row and first column gives the same value.
Reason: Expanding a determinant along any row or column gives the same value.

• A. Assertion is correct, reason is correct; reason is a correct explanation for assertion.
• B. Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• C. Assertion is correct, reason is incorrect
• D. Assertion is incorrect, reason is correct.

Answer 1

Answer: (A)

Expansion along second row $\left(R_2\right)$
$\left|A\right|=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
By expanding along $R_2$, we get
$\left|A\right|={\left(-1\right)}^{2+1}{a}_{21}\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|+{\left(-1\right)}^{2+2}{a}_{22}\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|$
$+{\left(-1\right)}^{2+3}{a}_{23}\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|$
(applying all steps together\right)
$=-a_{21}\left(a_{12}{a}_{33}-a_{32}{a}_{13}\right)+a_{22}\left(a_{11}{a}_{33}-a_{31}{a}_{13}\right)-a_{23}\left(a_{11}{a}_{22}-a_{21}{a}_{12}\right)$
$|A|=-a_{21}{a}_{12}{a}_{33}+a_{21}{a}_{32}{a}_{13}+a_{22}{a}_{11}{a}_{33}$
$-a_{32}{a}_{31}{a}_{13}-a_{33}{a}_{11}{a}_{32}+a_{33}{a}_{31}{a}_{12}$
$=a_{11}{a}_{22}{a}_{33}=a_{11}{a}_{23}{a}_{32}=a_{12}{a}_{21}{a}_{33}\mid a_{12}{a}_{23}{a}_{31}$
$+a_{13}{a}_{21}{a}_{32}-a_{13}{a}_{31}{a}_{22}\dots (i)$
Expansion along first column $(C_1)$
$\left|A\right|=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
By expanding along $C_1$, we get
$\left|A\right|=a_{11}{\left(-1\right)}^{1+1}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|+a_{21}{\left(-1\right)}^{2+1}\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|+a_{31}{\left(-1\right)}^{3+1}\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{22}& a_{23}\end{array}\right|$
$=a_{11}\left(a_{22}{a}_{33}-a_{23}{a}_{32}\right)-a_{21}\left(a_{12}{a}_{3y}-a_{13}{a}_{32}\right)$
$|A|=a_{11}{a}_{22}{a}_{33}-a_{11}{a}_{23}{a}_{32}-a_{21}{a}_{12}{a}_3$
$+a_{21}{a}_{13}{a}_{32}+a_{31}{a}_{12}{a}_{23}-a_{31}{a}_{13}{a}_{22}$
$=a_{11}{a}_{22}{a}_{33}-a_{11}{a}_{23}{a}_{32}-a_{12}{a}_{21}{a}_{33}+a_{12}{a}_{23}{a}_{31}<br/><br/>+a_{13}{a}_{21}{a}_{32}-a_{13}{a}_{31}{a}_{22}\dots (ii)$
Clearly, the values of $|A|$ in eqs. (i) and (ii) are equal. Also, it can be easily verified that the values of $|A|$ by expanding along $R_3,C_2$ and $C_3$ are equal to the value of $|A|$ obtained in eqs. (i) and (ii).
Hence, expanding a determinant along any row or column gives same value.