Tag: minors and cofactors

Question 1

Consider the determinant $\Delta=\begin{vmatrix}a _1 & a _2 & a _3 \\ b _1 & b _2 & b _3 \\ c _1 & c _2 & c _3\end{vmatrix}$

$M _{ij} =$ Minor of the element of $i^{th}$ row & $j^{th}$ column.

$C _{ij} =$ Cofactor of element of $i^{th}$ row & $j^{th}$ column.

$a _2.C _{12} + b _2.C _{22} + c _2.C _{32}$ is equal to

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$0$
$\Delta$
$2\Delta$
$\Delta^2$

Answer

Correct Option: B

Question 2

Consider the determinant $\Delta=\begin{vmatrix}a _1 & a _2 & a _3 \\ b _1 & b _2 & b _3 \\ c _1 & c _2 & c _3\end{vmatrix}$

$M _{ij} =$ Minor of the element of $i^{th}$ row & $j^{th}$ column.

$C _{ij} =$ Cofactor of element of $i^{th}$ row & $j^{th}$ column.

Value of $b _1.C _{31} + b _2.C _{32} + b _3.C _{33}$ is

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$0$
$\Delta$
$2\Delta$
$\Delta^2$

Answer

Correct Option: A

Question 3

$A,B,C$ are cofactors of elements, $\mathrm{a},\ \mathrm{b},\ \mathrm{c}$ in

${\begin{bmatrix}
a & b & c\
2 & 4 & 7\
-1 & 0 & 3
\end{bmatrix}}$ then the value of $(2\mathrm{A}+4\mathrm{B}+7\mathrm{C})$
is equal to

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$0$
2
$-1$
4

Answer

Correct Option: A

Question 4

If $\displaystyle A=\left[ { a } _{ ij } \right] $ is a $4 \times 4$ matrix and $\displaystyle { c } _{ ij }$ is the co-factor of the element $\displaystyle { a } _{ ij }$ in $\displaystyle \left| A \right| $, then the expression $\displaystyle { a } _{ 11 }{ c } _{ 11 }+{ a } _{ 12 }{ c } _{ 12 }+{ a } _{ 13 }{ c } _{ 13 }+{ a } _{ 14 }{ c } _{ 14 }$ equals

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$0$
$-1$
$1$
$\displaystyle \left| A \right| $

Answer

Correct Option: D

Question 5

If in $\displaystyle \left[ \begin{matrix} { a } _{ 1 } \ { a } _{ 2 } \ { a } _{ 3 } \end{matrix}\begin{matrix} { b } _{ 1 } \ { b } _{ 2 } \ { b } _{ 3 } \end{matrix}\begin{matrix} { c } _{ 1 } \ { c } _{ 2 } \ { c } _{ 3 } \end{matrix} \right] $, the cofactor of $\displaystyle { a } _{ r }$ is $\displaystyle { A } _{ r }$, then $\displaystyle { c } _{ 1 }{ A } _{ 1 }+{ c } _{ 2 }{ A } _{ 2 }+{ c } _{ 3 }{ A } _{ 3 }$ is

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\displaystyle 0$
$\displaystyle -D$
$\displaystyle D$
$\displaystyle { D }^{ 2 }$

Answer

Correct Option: A

Question 6

If $A=\begin{bmatrix} 3 & 2 & 4 \ 1 & 2 & 1 \ 3 & 2 & 6 \end{bmatrix}$ and $A _{ij}$ are the cofactors of $a _{ij}$, then $a _{11}A _{11}+a _{12}A _{12}+a _{13}A _{13}$ is equal to

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$8$
$6$
$4$
$0$

Answer

Correct Option: A

Question 7

If ${A} _{1}, {B} _{1}, {C} _{1}..$ are respectively the co-factor of the elements ${a} _{1}, {b} _{1}, {c} _{1}$.
$\triangle =\begin{vmatrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \ a _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{vmatrix}$, then $\begin{vmatrix} { B } _{ 2 } & C _{ 2 } \ B _{ 3 } & C _{ 3 } \end{vmatrix}$

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

${a} _{1}\triangle$
${a} _{1}{a} _{3}\triangle$
$({a} _{1}+{b} _{1})\triangle$
$None\ of\ these$

Explanation:

$\begin{array}{l} { B _{ 2 } }={ a _{ 1 } }{ c _{ 3 } }-{ a _{ 3 } }{ c _{ 1 } } \ { c _{ 2 } }=-\left( { { a _{ 1 } }{ b _{ 3 } }-{ a _{ 3 } }{ b _{ 1 } } } \right) \ { B _{ 3 } }=-\left( { { a _{ 1 } }{ c _{ 2 } }-{ a _{ 2 } }{ c _{ 1 } } } \right) \ { c _{ 3 } }={ a _{ 1 } }{ b _{ 2 } }-{ b _{ 1 } }{ a _{ 2 } } \ \left| \begin{array}{l} { B _{ 2 } } & { C _{ 2 } } \ { B _{ 3 } } & { C _{ 3 } } \end{array} \right| =\left| \begin{array}{l} { a _{ 1 } }{ c _{ 3 } }-{ a _{ 3 } }{ c _{ 1 } } & -{ a _{ 1 } }{ b _{ 3 } }+{ a _{ 3 } }{ b _{ 1 } } \ -{ a _{ 1 } }{ c _{ 2 } }+{ a _{ 2 } }{ c _{ 1 } } & { a _{ 1 } }{ b _{ 2 } }-{ b _{ 1 } }{ a _{ 2 } } \end{array} \right| \ =\left| \begin{array}{l} { a _{ 1 } }{ c _{ 3 } } & -{ a _{ 1 } }{ b _{ 3 } } \ -{ a _{ 1 } }{ c _{ 2 } } & { a _{ 1 } }{ b _{ 2 } } \end{array} \right| +\left| \begin{array}{l} { a _{ 1 } }{ c _{ 3 } } & { a _{ 3 } }{ b _{ 1 } } \ -{ a _{ 1 } }{ c _{ 2 } } & -{ a _{ 2 } }{ b _{ 1 } } \end{array} \right| +\left| \begin{array}{l} -{ a _{ 3 } }{ c _{ 1 } } & -{ a _{ 1 } }{ b _{ 3 } } \ { a _{ 2 } }{ c _{ 1 } } & { a _{ 1 } }{ b _{ 2 } } \end{array} \right| \ 1\left| \begin{array}{l} -{ a _{ 3 } }{ c _{ 1 } } & { a _{ 3 } }{ b _{ 1 } } \ { a _{ 2 } }{ c _{ 1 } } & -{ a _{ 2 } }{ b _{ 1 } } \end{array} \right| \ =a _{ 1 }^{ 2 }\left| \begin{array}{l} { c _{ 3 } } & -{ b _{ 3 } } \ -{ c _{ 2 } } & { b _{ 2 } } \end{array} \right| +{ a _{ 1 } }{ b _{ 1 } }\left| \begin{array}{l} { c _{ 3 } } & { a _{ 3 } } \ -{ c _{ 2 } } & -{ a _{ 2 } } \end{array} \right| +{ a _{ 1 } }c\left| \begin{array}{l} -{ a _{ 3 } } & -{ b _{ 3 } } \ { a _{ 2 } } & { b _{ 2 } } \end{array} \right| +{ b _{ 1 } }{ c _{ 1 } }\left| \begin{array}{l} -{ a _{ 3 } } & { a _{ 3 } } \ { a _{ 2 } } & -{ a _{ 2 } } \end{array} \right| \ ={ a _{ 1 } }\left{ { { a _{ 1 } }\left( { { b _{ 2 } }{ c _{ 3 } }-{ b _{ 3 } }{ c _{ 2 } } } \right) -{ b _{ 1 } }\left( { { a _{ 2 } }{ c _{ 3 } }-{ a _{ 3 } }{ c _{ 2 } } } \right) +{ c _{ 1 } }\left( { { a _{ 2 } }{ b _{ 3 } }-{ a _{ 3 } }{ b _{ 2 } } } \right) } \right} \ ={ a _{ 1 } }\left| \begin{array}{l} { a _{ 1 } } & { b _{ 1 } } & { c _{ 1 } } \ { a _{ 2 } } & { b _{ 2 } } & { c _{ 2 } } \ { a _{ 3 } } & { b _{ 3 } } & { c _{ 3 } } \end{array} \right| \ ={ a _{ 1 } }\Delta \end{array}$

Answer

Correct Option: A

Question 8

If $\Delta =\left| \begin{matrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \ { a } _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{matrix} \right|$ and $A _{1},B _{1},C _{1}$ denote the co-factors of $a _{1},b _{2},c _{1}$ respectively, then the value of the determinant $\left| \begin{matrix} { A } _{ 1 } & { B } _{ 1 } & { C } _{ 1 } \ { A } _{ 2 } & { B } _{ 2 } & { C } _{ 2 } \ { A } _{ 3 } & { B } _{ 3 } & { C } _{ 3 } \end{matrix} \right|$ is

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\Delta$
$\Delta^{2}$
$\Delta^{3}$
$0$

Answer

Correct Option: B

Question 9

If $\Delta =\begin{vmatrix} a _{11} & a _{12} & a _{13}\ a _{21} & a _{22} & a _{23}\ a _{31} & a _{32} & a _{33} \end{vmatrix}$ and $c _{ij}=\left ( -1 \right )^{i+j}$ (determinant obtained by deleting ith row and jth column), then $\begin{vmatrix} c _{11} & c _{12} & c _{13}\ c _{21} & c _{22} & c _{23}\ c _{31} & c _{32} & c _{33} \end{vmatrix}=\Delta ^{2}$

If $\begin{vmatrix} 1 & x & x^{ 2 } \ x & x^{ 2 } & 1 \ x^{ 2 } & 1 & x \end{vmatrix}=7$ and $\Delta =\begin{vmatrix}
x^{3}-1 & 0 & x-x^{4}\
0 & x-x^{4} & x^{3}-1\
x-x^{4} & x^{3}-1 & 0
\end{vmatrix}$, then

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\Delta =7$
$\Delta =343$
$\Delta =-49$
$\Delta =49$

Answer

Correct Option: D

Question 10

Let $\Delta _0=\begin{bmatrix}a _{11} & a _{12} & a _{13}\a _{21} & a _{22} &a _{23} \ a _{31} & a _{32} & a _{33}\end{bmatrix}$ (where $\Delta _0 \neq 0$) and let $\Delta _1$ denote the determinant formed by the cofactors of elements of $\Delta _0$ and $\Delta _2$ denote the determinant formed by the cofactor at $\Delta _1$ and so on $\Delta _n$ denotes the determinant formed by the cofactors at $\Delta _{n-1}$ then the determinant value of $\Delta _{n}$ is

mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\Delta _0^{2n}$
$\Delta _0^{2^n}$
$\Delta _0^{n^2}$
$\Delta _0^{2}$

Answer

Correct Option: B

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