Paragraph: Let Δ be the determinant of a matrix A , where A= ( array lll a_ 11 & a_ 12 & a_ 13 a_ 21 & a_ 22 & a_ 23 a_ 31 & a_ 32 & a_ 33 array ) and C_ 11 , C_ 12 , C_ 13 be the cofactors of a_ 11 , a_ 12 , a_ 13

Correct answer: D. -Δ Δ=| array lll a_ 11 & a_ 12 & a_ 13 a_ 21 & a_ 22 & a_ 23 a_ 31 & a_ 32 & a_ 33 array |=| array lll a_ 11 & a_ 21 & a_ 31 a_ 12 & a_ 22 & a_ 32 a_ 13 & a_ 23 & a_ 33 array | Apply C_1 C_3 Apply C_1 C_2=-| array lll a_ 31 & a_ 21 & a_ 11 a_ 32 & a_ 22 & a_ 12 a_ 33 & a_ 23 & a_ 13 array |=+| array lll a_ 21 & a_ 31 & a_ 11 a_ 22 & a_ 32 & a_ 12 a_ 23 & a_ 33 & a_ 13 array | Apply R_2 R_3=-| array lll a_ 21 & a_ 31 & a_ 11 a_ 23 & a_ 33 & a_ 13 a_ 22 & a_ 32 & a_ 12 array |

NDA 2022

Exam: NDA
Year: 2022
Subject: Mathematics
Chapter: Determinants

NDA 2022 · Mathematics · Determinants

Paragraph: Let Δ be the determinant of a matrix A , where A= (...

Paragraph: Let

\Delta

be the determinant of a matrix

A

, where

A=

\left(

\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}

\right)

and

C_{11}, C_{12}, C_{13}

be the cofactors of

a_{11}, a_{12}, a_{13}

respectively.<br><strong>Question:</strong> What is the value of

\left|

\begin{array}{lll}a_{21} & a_{31} & a_{11} \\ a_{23} & a_{33} & a_{13} \\ a_{22} & a_{32} & a_{12}\end{array}

\right| ?

Options

A $0$
B $1$
C $\Delta$
D $-\Delta$

Correct answer

D. $-\Delta$

Step-by-step solution

\Delta=\left|

\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}

\right|=\left|

\begin{array}{lll} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{array}

\right|

Apply

C_1 \longleftrightarrow C_3

Apply

C_1 \longleftrightarrow C_2=-\left|

\begin{array}{lll} a_{31} & a_{21} & a_{11} \\ a_{32} & a_{22} & a_{12} \\ a_{33} & a_{23} & a_{13} \end{array}

\right|=+\left|

\begin{array}{lll} a_{21} & a_{31} & a_{11} \\ a_{22} & a_{32} & a_{12} \\ a_{23} & a_{33} & a_{13} \end{array}

\right|

Apply

R_2 \longleftrightarrow R_3=-\left|

\begin{array}{lll} a_{21} & a_{31} & a_{11} \\ a_{23} & a_{33} & a_{13} \\ a_{22} & a_{32} & a_{12} \end{array}

\right|

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