Tag: matrices and determinants

Questions Related to matrices and determinants

$\begin{vmatrix}a^2 + x^2 & ab - cx & ac + bx\ ab+ cx & b^2 + x^2 & bc - ax\ ac - bx & bc + ax & c^2 + x^2\end{vmatrix} =$

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

  1. $ \begin{vmatrix}x & b & -c\ -a & x & c\ a & -b & x\end{vmatrix}^2$

  2. $ \begin{vmatrix}x & -b & c\ a & x & -c\ -a & b & x\end{vmatrix}^2$

  3. $ \begin{vmatrix}x & c & -b\ -c & x & a\ b & -a & x\end{vmatrix}^2$

  4. $ \begin{vmatrix}x & -c & b\ c & x & -a\ -b & a & x\end{vmatrix}^2$

Show answer
Correct Option: C,D
Explanation:

Let $D = \begin{vmatrix} x & c & -b\ -c & x & a\ b & -a & x\end{vmatrix}$
Cofactors of 1st Row of D are
$x^2 + a^2 ,  ab + cx, ac - bx$
Cofactor of 2nd Row of D are
$ab - cx,x^2 + b^2, ax + bc$
and cofactors of 3rd row of D are
$ax + bx, bc - ax, x^2 + c^2$
$\therefore $ Determinant of cofactors of D is
$D^c

= \begin{vmatrix}x^2 +a^2 & ab + cx & ac - bx\ ab - cx &

x^2 + b^2 & ax + bc\ ac + bx & bc - ax & x^2 +

c^2\end{vmatrix}$
$= \begin{vmatrix}a^2 + x^2 & ab- cx & ac

+ bx\ ab + cx & b^2 + x^2 & bc - ax\ ac - bx & ax +

bc & c^2 + x^2\end{vmatrix}$              (Rows interchanging into columns)
$= D^2$
$= \begin{vmatrix} x & c &-b^2 \ -c

& x & a\ b & -a & x\end{vmatrix}^2$            

($\because D^c = D^2$, D is third order determinant)
Hence,
$\begin{vmatrix}a^2

+ x^2 & ab - cx & ac + bx\ ab+cx & b^2 + x^2 & bc -

ax\ ac - bx & ax + bc &c^2 + x^2 \end{vmatrix}

=  \begin{vmatrix}x & c & -b\ -c & x & a\ b & -a

& x\end{vmatrix}^2$

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 $A _1, B _1, C _1$ denote the co-factors of $a _1, b _1, c _1$ respectively, then the value of the determinant $\begin{vmatrix}A _1 & B _1 & C _1\A _2 & B _2 & C _2\ A _3 & B _3 & C _3\end{vmatrix}$ is

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

  1. $\Delta$

  2. $\Delta^2$

  3. $\Delta^3$

  4. $0$

Show answer
Correct Option: A

If $A=\begin{bmatrix} a & c & b\ b & a & c\ c & b & a\end{bmatrix}$ then the cofactor of $a _{32}$ in $A+A^T$ is?

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

  1. $-(2a(b+c)-(b+c)^2)$

  2. $ac-b^2$

  3. $a^2-bc$

  4. $2a(a+c)-(a+c)^2$

Show answer
Correct Option: A

$\displaystyle A _{1},B _{1},C _{1}$ 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

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

  1. $\displaystyle a _{1}a _{3}\Delta $

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

  3. $\displaystyle a _{1} \Delta $

  4. None of these

Show answer
Correct Option: C
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 ${ a } _{ 1 }\Delta =\begin{vmatrix} { B } _{ 2 } & { C } _{ 2 } \ { B } _{ 2 } & { C } _{ 3 } \end{vmatrix}$

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

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

  1. $-\Delta$

  2. 0

  3. $\Delta$

  4. none of these

Show answer
Correct Option: B
Explanation:

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

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

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

  1. $0$

  2. $-1$

  3. $1$

  4. $Det. (A)$

Show answer
Correct Option: D

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

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

  1. $|C|=|A|$

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

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

  4. none of these

Show answer
Correct Option: B
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

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

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

  1. $7 + 4i$

  2. $7 - 4i$

  3. $4 + 7i$

  4. $4 - 7i$

Show answer
Correct Option: C
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$

 $=4+7i$

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

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

  1. $13$

  2. $-13$

  3. $5$

  4. $-5$

Show answer
Correct Option: A
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

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

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

  1. $0$

  2. $4\Delta$

  3. $2\Delta$

  4. $\Delta$

Show answer
Correct Option: D
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 ${ C } _{ 3 }$