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 $\left( R _{2}\right)$
$\left|A\right|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix}$
By expanding along $R_{2}$, we get
$\left|A\right|=\left(-1\right)^{2+1} a_{21} \begin{vmatrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{vmatrix}+\left(-1\right)^{2+2} a_{22} \begin{vmatrix}a_{11}&a_{13}\\ a_{31}&a_{33}\end{vmatrix}$
$+\left(-1\right)^{2+3} a_{23} \begin{vmatrix}a_{11}&a_{12}\\ a_{31}&a_{32}\end{vmatrix} $
(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 $(C_{1})$
$\left|A\right|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix}$
By expanding along $C_{1}$, we get
$\left|A\right|=a_{11}\left(-1\right)^{1+1} \begin{vmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{vmatrix}+a_{21} \left(-1\right)^{2+1}\begin{vmatrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{vmatrix}+a_{31}\left(-1\right)^{3+1}\begin{vmatrix}a_{12}&a_{13}\\ a_{22}&a_{23}\end{vmatrix}$
$=a_{11}\left(a_{22} a_{33}-a_{23} a_{32}\right)-a_{21}\left(a_{12} a_{3 y}-a_{13} a_{32}\right)$
$| A |= a _{11} a _{22} a _{33}- a _{11} a _{23} a _{32}- a _{21} a _{12} a _{3}$
$+ a _{21} a _{13} a _{32}+ a _{31} a _{12} a _{23}- a _{31} a _{13} a _{22}$
$=a_{11} a_{22} a_{33}-a_{11} a_{23} a_{32}-a_{12} a_{21} a_{33}+a_{12} a_{23} a_{31}
+a_{13} a_{21} a_{32}-a_{13} a_{31} a_{22} \,\,\,\, \dots(ii)$
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.