Minors and cofactors - class-XII

$\Delta =\begin{vmatrix} { a } _{ 11 } & { a } _{ 12 } & { a } _{ 13 } \ { a } _{ 21 } & { a } _{ 22 } & { a } _{ 23 } \ { a } _{ 31 } & { a } _{ 32 } & { a } _{ 33 } \end{vmatrix}$

By property of cofactor (2),
$A=\begin{bmatrix}
8 & 9\
10 & 11
\end{bmatrix}$
minor of $a _{12}=M _{12}=10$
So, cofactor of $a _{12}=M _{12}(-1)^{1+2}=-M _{12}=-10$

Co-factor of $ \displaystyle a _2 = (-1)^{i+j}\left|\begin{matrix} b _1 & c _1 \ b _3 & c _3 \end{matrix}\right| = - [b _1 , c _3 - c _1 , b _3] = A _2 $

Given,

From the property of determinants, if any element is multiplied with cofactor of corresponding element of another row, and summed up for each element of original row, the sum comes out zero $\Rightarrow (A)$

The element 4 is located at the second row, third column (a_23). Its minor is obtained by deleting the second row and third column: det([[1, 3, 1], [8, 0, 1], [0, 2, 1]]). Evaluating this determinant gives 1(0 - 2) - 3(8 - 0) + 1(16 - 0) = -2 - 24 + 16 = -10. The cofactor is (-1)^(2+3) * minor = -1 * (-10) = 10.

Let $D=\begin{vmatrix} \lambda  & c & -b \ -c & \lambda  & a \ b & -a & \lambda  \end{vmatrix}$

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$

The determinant formed by the cofactors of the elements of a matrix is equal to the square of the original determinant, i.e., Delta^2, a standard property of determinants involving adjoint matrices.

A + A^T is a symmetric matrix. The element a32 of A+A^T is the sum of the (3,2) element of A and A^T, which is b + c. The cofactor of this element in the resulting matrix involves calculating the minor of the (3,2) position.

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 ${ a } _{ 1 }\Delta =\begin{vmatrix} { B } _{ 2 } & { C } _{ 2 } \ { B } _{ 2 } & { C } _{ 3 } \end{vmatrix}$

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

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

$C\rightarrow$ Cofactor matrix

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

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

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

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

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

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

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

${ 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

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

$\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}$

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

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

$\Delta _1=\Delta ^2 _0,\Delta _2=\Delta ^2 _1=\Delta ^{2^2} _0$
$\therefore \Delta _n=\Delta ^{2n} _0$