Mathematics

Linear Algebra

449 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\displaystyle A _{1},B _{1},C _{1}$ are respectively the co-factors of $\displaystyle a _{1},b _{1},c _{1}$ of the determinant $\displaystyle \Delta = \begin{vmatrix}a _{1} &b _{1}  &c _{1} \a _{2}  &b _{2}  &c _{2} \a _{3} &b _{3}  &c _{3}\end{vmatrix}$ then $\displaystyle \begin{vmatrix}B _{2} &C _{2} \B _{3} &C _{3}\end{vmatrix}$ equals

  1. $\displaystyle a _{1}a _{3}\Delta $
  2. $\displaystyle (a _{1}-b _{1})\Delta $
  3. $\displaystyle a _{1} \Delta $

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For $\Delta =\begin{vmatrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \ { a } _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{vmatrix}$ 
Let $R=\begin{vmatrix} { A } _{ 1 } & { B } _{ 1 } & { C } _{ 1 } \ { A } _{ 2 } & { B } _{ 2 } & { C } _{ 2 } \ { A } _{ 3 } & { B } _{ 2 } & { C } _{ 3 } \end{vmatrix}$ is the matrix of cofactor 
Then ${ a } _{ 1 }\Delta =\begin{vmatrix} { B } _{ 2 } & { C } _{ 2 } \ { B } _{ 2 } & { C } _{ 3 } \end{vmatrix}$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

If $\Delta =\begin{vmatrix} a _1 & b _1 & c _1 \ a _2 & b _2 & c _2 \ a _3 & b _3 & c _3\end{vmatrix}$ and $A _2, B _2, C _2$ are respectively cofactors of $a _2, b _2, c _2$ then $a _1A _2 + b _1B _2 + c _1C _2$ is equal to

  1. $-\Delta$

  2. 0
  3. $\Delta$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Delta =\begin{vmatrix} { a } _{ 1 } & { b } _{ 1 } & { c } _{ 1 } \ { a } _{ 2 } & { b } _{ 2 } & { c } _{ 2 } \ { a } _{ 3 } & { b } _{ 3 } & { c } _{ 3 } \end{vmatrix}\ { A } _{ 2 }=-\begin{vmatrix} { b } _{ 1 } & { c } _{ 1 } \ { b } _{ 3 } & { c } _{ 3 } \end{vmatrix}={ b } _{ 3 }{ c } _{ 1 }-{ b } _{ 1 }{ c } _{ 3 }\ B _{ 2 }=\begin{vmatrix} { a } _{ 1 } & { c } _{ 1 } \ { a } _{ 3 } & { c } _{ 3 } \end{vmatrix}={ c } _{ 3 }{ a } _{ 1 }-{ c } _{ 1 }{ a } _{ 3 }\ C _{ 3 }=-\begin{vmatrix} { a } _{ 1 } & { b } _{ 1 } \ { a } _{ 3 } & { b } _{ 3 } \end{vmatrix}={ a } _{ 3 }{ b } _{ 1 }-{ a } _{ 1 }{ b } _{ 3 }\ \therefore { a } _{ 1 }{ A } _{ 2 }+{ b } _{ 1 }B _{ 2 }+{ c } _{ 1 }C _{ 3 }={ a } _{ 1 }{ b } _{ 3 }{ c } _{ 1 }-{ a } _{ 1 }{ b } _{ 1 }{ c } _{ 3 }+{ a } _{ 1 }{ b } _{ 1 }{ c } _{ 3 }-{ a } _{ 3 }{ b } _{ 1 }{ c } _{ 1 }+{ a } _{ 3 }{ b } _{ 1 }{ c } _{ 1 }-{ a } _{ 1 }{ b } _{ 3 }{ c } _{ 1 }=0$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

If $A = (a _{ij})$ is a $4\times 4$ matrix and $C _{ij}$ is the co-factor of the element $a _{ij}$ in Det (A), then the expression $a _{11}C _{11} + a _{12}C _{12} + a _{13}C _{13} + a _{14}C _{14}$ equals

  1. $0$
  2. $-1$
  3. $1$
  4. $Det. (A)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The sum of the products of elements of any row and their corresponding cofactors is equal to the determinant of the matrix.

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

Let $A = [a _{ij}] _{n\times n}$ be a square matirx and let $c _{ij}$ be cofactor of $a _{ij}$ in A. If $C = [c _{ij}]$, then

  1. $|C|=|A|$
  2. $|C|=|A|^{n-1}$
  3. $|C|=|A|^{n-2}$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$C\rightarrow$ Cofactor matrix

$AdjA= \left (C \right )^{^{T}}$
But det of $AdjA= Det \quad of  \quad C$
Because they are transpore of each other .
$\Rightarrow\left  | AdjA \right | = \left | C \right |= \left | A \right |^{n-1} $
Option-B

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$\begin{vmatrix}1+i & 1-i & i \ 1-i & i & 1+i\ i & 1+i & 1-i\end{vmatrix}$ (where $i=\sqrt {-1}$ ) equals

  1. $7 + 4i$
  2. $7 - 4i$
  3. $4 + 7i$
  4. $4 - 7i$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\begin{vmatrix} 1+i & 1-i & i \ 1-i & i & 1+i \ i & 1+i & 1-i \end{vmatrix}\$


$ =\left( 1+i \right) \begin{vmatrix} i & 1+i \ 1+i & 1-i \end{vmatrix}-\left( 1-i \right) \begin{vmatrix} 1-i & 1+i \ i & 1-i \end{vmatrix}+i\begin{vmatrix} 1-i & i \ i & 1+i \end{vmatrix}\$

$ =\left( 1+i \right) \left( i+1-\left( 1-1+2i \right)  \right) -\left( 1-i \right) \left( 1-1-2i-i+1 \right) +i\left( 1+1+1 \right) \ $

$=\left( 1+i \right) \left( 1-i \right) -\left( 1-i \right) \left( 1-3i \right) +3i\$

$ =1+1-1+3+3i+i+3i$

 $=4+7i$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

If $A=\begin{bmatrix} 1 & -2 & 3 \ 4 & 0 & -1 \ -3 & 1 & 5 \end{bmatrix}$, then ${(adj. A)} _{23}$ is equal to

  1. $13$
  2. $-13$
  3. $5$
  4. $-5$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A=\begin{bmatrix} 1 & -2 & 3 \ 4 & 0 & -1 \ -3 & 1 & 5 \end{bmatrix}$


${(adj. A)} _{23}={C} _{32}$

So cofactor of ${a} _{32}$


${C} _{32}={(-1)}^{3+2}(-1-12)=13$

Ans: A

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants
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.

$a _3M _{13} - b _3M _{23} + c _3M _{33}$ is equal to

  1. $0$
  2. $4\Delta$
  3. $2\Delta$
  4. $\Delta$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

${ a } _{ 3 }{ M } _{ 13 }-{ b } _{ 2 }{ M } _{ 23 }+{ c } _{ 3 }{ M } _{ 33 }$


$ ={ a } _{ 3 }\begin{vmatrix} { b } _{ 1 }\quad  & { b } _{ 2 } \ { c } _{ 1 } & { c } _{ 2 } \end{vmatrix}-{ b } _{ 3 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 2 } \ { c } _{ 1 } & { c } _{ 2 } \end{vmatrix}+{ c } _{ 3 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 2 } \ { b } _{ 1 } & { b } _{ 2 } \end{vmatrix}$

Is equal to the expansion of $\triangle $ along ${ C } _{ 3 }$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants
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

  1. $0$
  2. $\Delta$
  3. $2\Delta$
  4. $\Delta^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The value of ${ a } _{ 2 }.{ C } _{ 12 }+{ b } _{ 2 }.{ C } _{ 22 }+{ C } _{ 2 }.{ C } _{ 32 }$


$ ={ a } _{ 2 }{ \left( -1 \right)  }^{ 1+2 }\begin{vmatrix} { b } _{ 1 }\quad  & { b } _{ 3 } \ { c } _{ 1 } & { c } _{ 3 } \end{vmatrix}+{ b } _{ 2 }.{ \left( -1 \right)  }^{ 2+2 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { c } _{ 1 } & { a } _{ 3 } \end{vmatrix}+{ c } _{ 2 }.{ \left( -1 \right)  }^{ 3+2 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { b } _{ 1 } & { b } _{ 3 } \end{vmatrix}$

$ =-{ a } _{ 2 }\begin{vmatrix} { b } _{ 1 }\quad  & { b } _{ 3 } \ { c } _{ 1 } & { c } _{ 3 } \end{vmatrix}+{ b } _{ 2 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { c } _{ 1 } & { a } _{ 3 } \end{vmatrix}-{ c } _{ 2 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { b } _{ 1 } & { b } _{ 3 } \end{vmatrix}$

Is same as expansion of $\triangle $ along ${ C } _{ 2 }$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants
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

  1. $0$
  2. $\Delta$
  3. $2\Delta$
  4. $\Delta^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Value of ${ b } _{ 1 }.{ C } _{ 31 }+{ b } _{ 2 }.C _{ 32 }+b _{ 3 }.{ C } _{ 33 }$


$={ b } _{ 1 }.{ \left( -1 \right)  }^{ 3+1 }\begin{vmatrix} { a } _{ 2 }\quad  & { a } _{ 3 } \ { b } _{ 2 } & { b } _{ 3 } \end{vmatrix}+{ b } _{ 3 }.{ \left( -1 \right)  }^{ 3+2 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { b } _{ 1 } & { b } _{ 3 } \end{vmatrix}+{ b } _{ 3 }.{ \left( -1 \right)  }^{ 3+3 }\begin{vmatrix} { a } _{ 1 }\quad  & { a } _{ 3 } \ { b } _{ 1 } & { b } _{ 2 } \end{vmatrix}$


$={ b } _{ 1 }\left( { b } _{ 2 }{ a } _{ 3 }-{ a } _{ 2 }{ b } _{ 3 } \right) -{ b } _{ 2 }\left( { b } _{ 1 }{ a } _{ 3 }-{ a } _{ 1 }{ b } _{ 3 } \right) +{ b } _{ 3 }\left( { b } _{ 1 }{ a } _{ 3 }-{ a } _{ 1 }{ b } _{ 2 } \right)$

$=0$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

$A,B,C$ 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

  1. $0$

  2. 2
  3. $-1$

  4. 4
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A = 4\times 3-0\times 7 = 12$
$B = -(2\times 3-7\times (-1)) = -13$
$C = 2\times 0-4\times (-1) = 4$
$2A+4B+7C = 0$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

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

  1. $0$
  2. $-1$
  3. $1$
  4. $\displaystyle \left| A \right| $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$A=\left[ { a } _{ ij } \right] \quad 4\times 4$

${ c } _{ ij }\rightarrow $co factor

$A=\begin{bmatrix} { a } _{ 11 } & { a } _{ 12 } & { a } _{ 13 } \\ { a } _{ 21 } & { a } _{ 22 } & { a } _{ 23 } \\ { a } _{ 31 } & { a } _{ 32 } & { a } _{ 33 } \end{bmatrix}$ co factor $=$ Minor $\times \begin{matrix} + & - & + \\ - & + & - \\ + & - & + \end{matrix}$

Co factor matrix$=\begin{bmatrix} { a } _{ 22 }{ a } _{ 33 }-{ a } _{ 23 }{ a } _{ 32 } & -{ a } _{ 21 }{ a } _{ 33 }+{ a } _{ 23 }{ a } _{ 31 } & { a } _{ 21 }{ a } _{ 32 }-{ a } _{ 22 }{ a } _{ 31 } \\ { -a } _{ 12 }{ a } _{ 33 }+{ a } _{ 13 }{ a } _{ 32 } & { a } _{ 11 }{ a } _{ 33 }-{ a } _{ 13 }{ a } _{ 31 } & -{ a } _{ 11 }{ a } _{ 32 }+{ a } _{ 12 }{ a } _{ 31 } \\ { a } _{ 12 }{ a } _{ 23 }-{ a } _{ 13 }{ a } _{ 22 } & -{ a } _{ 11 }{ a } _{ 23 }+{ a } _{ 13 }{ a } _{ 21 } & { a } _{ 11 }{ a } _{ 22 }-{ a } _{ 12 }{ a } _{ 21 } \end{bmatrix}$

${ c } _{ 11 }={ a } _{ 22 }{ a } _{ 33 }-{ a } _{ 23 }{ a } _{ 32 }$

${ c } _{ 12 }={ a } _{ 23 }{ a } _{ 31 }-{ a } _{ 21 }{ a } _{ 33 }$

${ c } _{ 13 }={ a } _{ 21 }{ a } _{ 32 }-{ a } _{ 22 }{ a } _{ 31 }$

${ a } _{ 11 }{ c } _{ 11 }+{ a } _{ 12 }{ c } _{ 12 }+{ a } _{ 13 }{ c } _{ 13 }$

$=\left| A \right| $

Parallelly For $4\times 4$ matrix

Also

${ a } _{ 11 }{ c } _{ 11 }+{ a } _{ 12 }{ c } _{ 12 }+{ a } _{ 13 }{ c } _{ 13 }+{ a } _{ 14 }{ c } _{ 14 }=\left| A \right| $

Option D

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

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

  1. $8$
  2. $6$
  3. $4$
  4. $0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$a _{11}A _{11}+a _{12}A _{12}+A _{13}A _{13}$
$=3\begin{vmatrix} 2 & 1\ 2 & 6\end{vmatrix} -2\begin{vmatrix} 1 & 1\3 & 6\end{vmatrix} +4\begin{vmatrix} 1 &2 \ 3 & 2 \end{vmatrix}$
$=3(12-2)-2(6-3)+4(2-6)$
$=30-6-16$
$=8$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

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 $\begin{vmatrix} { B } _{ 2 } & C _{ 2 } \ B _{ 3 } & C _{ 3 } \end{vmatrix}$

  1. ${a} _{1}\triangle$
  2. ${a} _{1}{a} _{3}\triangle$
  3. $({a} _{1}+{b} _{1})\triangle$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
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}$

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

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

  1. $\Delta$
  2. $\Delta^{2}$
  3. $\Delta^{3}$
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The determinant of the matrix formed by cofactors of a 3x3 matrix is equal to the square of the original determinant (Delta^2).

Multiple choice mathematics and statistics determinants minors and cofactors determinants and matrices matrices and determinants

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

  1. $\Delta =7$
  2. $\Delta =343$
  3. $\Delta =-49$
  4. $\Delta =49$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For $\begin{vmatrix} 1 & x & x^{ 2 } \ x & x^{ 2 } & 1 \ x^{ 2 } & 1 & x \end{vmatrix}=7$
$\begin{vmatrix} c _{ 11 } & c _{ 12 } & c _{ 13 } \ c _{ 21 } & c _{ 22 } & c _{ 23 } \ c _{ 31 } & c _{ 32 } & c _{ 33 } \end{vmatrix}=\begin{vmatrix} x^{ 3 }-1 & 0 & x-x^{ 4 } \ 0 & x-x^{ 4 } & x^{ 3 }-1 \ x-x^{ 4 } & x^{ 3 }-1 & 0 \end{vmatrix}$
$\Delta ={ 7 }^{ 2 }=49$