Eigen Value Problem

Let $A$ be a square matrix of order $n$. If there exists a scalar $\lambda$ and an non-zero vector such that $AX=$ $\lambda X,\lambda$ is called an eigen value of $A\mathrm{\&}X$, an eigen vector of $A$.

The homogeneous system

$(A-\lambda I)X=0$ has non trivial solutions of $|A-\lambda I|=0$

which gives a polynomial of degree $n$. This equation is called the characteristic equation of $A$ and its roots are eigen values.

Properties:

• The sum of all eigen values $=$ trace of $A=\sum _{\mathrm{I}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{ii}}$
• Product of all eigen values $=|\mathrm{A}|$
• If $A$ is singular, then one of the eigen values is zero
• The eigen value of a null matrix is zero
• The eigen value of a unit matrix is unity.
• All the eigen values of a real orthogonal matrix have unit modulus i.e. $|\lambda |=1$

Cayley-Hamilton Theorem

Every square matrix A satisfies its characteristic equation $|A-\lambda I|=0$

Solution of a system of linear equation (matrix method)

Solution of the system ${\mathrm{a}}_{11}\mathrm{x}+{\mathrm{a}}_{12}\mathrm{y}+{\mathrm{a}}_{13}\mathrm{z}={\mathrm{b}}_1$

$\begin{array}{rl}& a_{21}x+a_{22}y+a_{23}z=b_2\\ & a_{31}x+a_{32}y+a_{33}z=b_3\end{array}\text{ is given }$

by $X=A^{-1}B$ where $A=\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right],B=\left[\begin{array}{l}{b}_1\\ b_2\\ b_3\end{array}\right]$

If $|\mathrm{A}|\ne 0$, the system is consistent and have unique solution.

If $|A|=0$, find $(\mathrm{adj}A)B$.

If $(\mathrm{adj}A)B=0$, the system has infinitely many solutions.

If (adj A) $B\ne 0$ the system is inconsistent and has no solution.

Echelon from of a matrix

A matrix A is said to be in echelon form if

i. The first non-zero element in each row is 1

ii. Every non-zero row in A precedes every zero row

iii. The number of zeroes before first non-zero element in $1^{\text{st }},2^{\text{nd }},3^{\text{rd }},\dots$ rows should be in increasing order.

Rank of a matrix

A non zero matrix ${\mathrm{A}}_{\mathrm{mxn}}$ is said to have rank $\mathrm{r}$ if at least one of its $\mathrm{rxr}$ minors is nonzero while every $(\mathrm{r}+1)$ $x(r+1)$ minor, if any, is zero. It is denoted as $\rho (A)=r$

Rank of a null matrix is zero.

Important results

$\rho (\mathrm{A})=\le min\mathrm{m},\mathrm{n}$

$\rho \left({\mathrm{I}}_{\mathrm{n}}\right)=\mathrm{n}$

If $\mathrm{A}$ is nonsingular, $\rho (A)=\mathrm{n}$ otherwise $\rho (A)<\mathrm{n}$.

Elementary operations do not change the rank of a matrix

Nullity of a matrix

For a square matrix of order $n,n-\rho (A)=n(A)$ is called the nullity of $A$ and is denoted by $N(A)$

Solution of a system by Rank method

Consider the non homogeneous system

$\left[\begin{array}{lll}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left|\begin{array}{l}{b}_1\\ b_2\\ b_2\end{array}\right|,AX=B$

Let $r$ be the rank of coefficient matrix and $s$ be the rank of the augmented matrix

$[AB]=\left[\begin{array}{llll}{a}_{11}& a_{12}& a_{13}& b_1\\ a_{21}& a_{22}& a_{23}& b_2\\ a_{31}& a_{32}& a_{33}& b_3\end{array}\right]$

If $r=s$, the system is consistent (has one or more solutions)

If $\mathrm{r}=\mathrm{s}=3$, (no. of unknowns), then the system has unique solutions.

If $r=s=2$ (< the no. of unknowns), then the system is consistent and has infinitely many solutions.

If $\mathrm{r}\ne \mathrm{s}$, the system is inconsistent (no solution)

Note: consider the system of planes

${\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1\mathrm{z}={\mathrm{d}}_1$

${\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2\mathrm{z}={\mathrm{d}}_2$

$a_{3}x+b_{3}y+c_{3}z=d_3$

Let $r$ be the rank of coefficient matrix and $s$ be the rank of the augmented matrix.

If $\mathrm{r}=3,\mathrm{s}=3$, the planes meet at a single point.

If $\mathrm{r}=2,\mathrm{s}=2$, the planes intersect along a single straight line

If $\mathrm{r}=2,\mathrm{s}=3$, the planes from a triangular prism

If $\mathrm{r}=1,\mathrm{s}=2$, the planes are parallel

If $\mathrm{r}=1,\mathrm{s}=1$, the planes are coincident.

Geometrical transformations (Rotation & Reflexions)

i. Reflection in $\mathrm{x}$-axis.

Let ${\mathrm{P}}^{\prime }({\mathrm{x}}^{\prime },{\mathrm{y}}^{\prime })$ be the reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ on $\mathrm{x}$-axis

then ${\mathrm{x}}^{\prime }=\mathrm{x}\mathrm{\&}{\mathrm{y}}^{\prime }=-\mathrm{y}$

i.e. $x^{\prime }=1.x+0.y$

${\mathrm{y}}^{\prime }=0.\mathrm{x}+(-1)\mathrm{y}$

i.e. $\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore$ The matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ describes reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ in the $\mathrm{x}$-axis

ii. Reflection in the y-axis

Here ${\mathrm{x}}^{\prime }=-\mathrm{x}\mathrm{\&}{\mathrm{y}}^{\prime }=\mathrm{y}$

i.e. $x^{\prime }=(-1)x+0y$

${\mathrm{y}}^{\prime }=0\mathrm{x}+1\mathrm{y}$

$\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore \left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ describes reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ in the $\mathrm{y}$-axis

iii. Reflection through the origin

${\mathrm{x}}^{\prime }=(-1)\mathrm{x}+0\mathrm{y}$

${\mathrm{y}}^{\prime }=0\mathrm{x}+(-1)\mathrm{y}$

$\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore \left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$ describes reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ through the origin

iv. Reflection in the line $y=x$

${\mathrm{x}}^{\prime }=0.\mathrm{x}+1.\mathrm{y}$

${\mathrm{y}}^{\prime }=1.\mathrm{x}+0.\mathrm{y}$

$\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore \left[\begin{array}{ll}0& 1\\ 1& 0\end{array}\right]$ describes reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ in the line $\mathrm{y}=\mathrm{x}$

v. Reflection in the line $\mathrm{y}=\mathrm{x}\tan \theta ($ or $\mathrm{y}=\mathrm{mx})$

${\mathrm{x}}^{\prime }=\mathrm{x}\cos 2\theta +\mathrm{y}\sin 2\theta$

${\mathrm{y}}^{\prime }=\mathrm{x}\sin 2\theta +\mathrm{y}(-\cos 2\theta )(\because {\mathrm{O}}^{\prime }$ is the mid point of $\mathrm{P}\mathrm{\&}{\mathrm{P}}^{\prime })$

$\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore \left[\begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{1-{\mathrm{m}}^2}{1+{\mathrm{m}}^2}}& {\displaystyle \frac{2\mathrm{m}}{1+{\mathrm{m}}^2}}\\ {\displaystyle \frac{2\mathrm{m}}{1+{\mathrm{m}}^2}}& {\displaystyle \frac{-(1-{\mathrm{m}}^2)}{1+{\mathrm{m}}^2}}\end{array}\right]$ describe reflection of $\mathrm{P}(\mathrm{x},\mathrm{y})$ in the line

$\mathrm{y}=\mathrm{x}\tan \alpha .($ or $\mathrm{y}=\mathrm{mx})$

vi. Rotation through an angle $\theta$.

$\mathrm{OP}={\mathrm{OP}}^{\prime }=\mathrm{r}$

Let $\mathrm{OP}$ rotate through an angle $\theta$ in anticlockwise direction.

${\mathrm{x}}^{\prime }=\mathrm{x}\cos \theta -\mathrm{y}\sin \theta$

${\mathrm{y}}^{\prime }=\mathrm{x}\sin \theta +\mathrm{y}\cos \theta$

$\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]$

$\therefore \left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ describes a rotation of a line segment through an angle $\theta$.

Solved Examples:

1. If $\mathrm{A}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& -2& 4\end{array}\right]$ and $\mathrm{I}=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and ${\mathrm{A}}^{-1}={\displaystyle \frac{1}{6}}({\mathrm{A}}^2-\mathrm{cA}+\mathrm{dI})$, then $\mathrm{c}$ and $\mathrm{d}$ are

(a) $-6,-11$

(b) 6,11

(c) $-6,11$

(d) $6,-11$

2. If $\mathrm{A}=\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]$, then ${\mathrm{A}}^4-5{\mathrm{A}}^3-{\mathrm{A}}^2-4\mathrm{A}-\mathrm{I}=$

(a) 0

(b) I

(c) A

(d) $\mathrm{A}+\mathrm{I}$

3. If $\mathrm{A}=\left[\begin{array}{cc}-1& {\displaystyle \frac{3}{2}}\\ {\displaystyle \frac{-1}{2}}& {\displaystyle \frac{1}{2}}\end{array}\right]$, then ${\mathrm{A}}^3=$

(a) ${\displaystyle \frac{\mathrm{A}}{4}}$

(b) ${\displaystyle \frac{\mathrm{A}}{8}}$

(c) ${\displaystyle \frac{\mathrm{I}}{4}}$

(d) ${\displaystyle \frac{\mathrm{I}}{8}}$

4. The number of values of $k$ for which the system of equations $(k+1)x+8y=4k,kx+(k+3)y=3k-1$ has infinitely many solutions

(a) 0

(b) 1

(c) 2

(d) Infinite

5. If the system of equations $x-ky-z=0,kx-y-z=0$ and $x+y-z=0$ has a non-zero solution, then $k=$

(a) 0

(b) 1

(c) -1

(d) 2

6. If the system of equations $x+2ay+az=0,x+3by+bz=0$ and $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$ are in

(a) $\mathrm{AP}$

(b) $\mathrm{GP}$

(c) $\mathrm{HP}$

(d) none of these

7. The system of equations

$\alpha x+y+z=\alpha -1$

$x+\alpha y+z=\alpha -1$ has no solution if $\alpha =$

$\mathrm{x}+\mathrm{y}+\alpha \mathrm{z}=\alpha -1$

(a) -2 or 1

(b) -2

(c) 1

(d) -1

Exercise:

1. If $c<1$ and the system of equation $x+y-1=0,2x-y-c=0$ and $-bx+3by-c=0$ is consistent, then the possible real values of $b$ are

(a) $\mathrm{b}\in (-3,{\displaystyle \frac{3}{4}})$

(b) $\mathrm{b}\in (-{\displaystyle \frac{3}{2}},4)$

(c) $\mathrm{b}\in (-{\displaystyle \frac{3}{4}},3)$

(d) none of these

2. If $\mathrm{a},\mathrm{b},\mathrm{c}$ are in G.P with common ration ${\mathrm{r}}_1$ and $\alpha ,\beta ,\gamma$ are in G.P. with common ratio ${\mathrm{r}}_2$ and equations $ax+\alpha y+z=0,bx+\beta y+z=0,ax+\gamma y+z=0$ have only zero solution. Then which of the following is not true?

(a) ${\mathrm{r}}_1\ne 1$

(b) ${\mathrm{r}}_2\ne 1$

(c) $r_1\ne r_2$

(d) none of these

3. If $f(x)=a+bx+c{x}^2$ and $\alpha ,\beta ,\gamma$ are roots of the equation $x^3=1$, then $\left|\begin{array}{lll}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is equal to

(a) $\mathrm{f}(\alpha )+\mathrm{f}(\beta )+\mathrm{f}(\gamma )$

(b) $\mathrm{f}(\alpha )\mathrm{f}(\beta )+\mathrm{f}(\beta )\mathrm{f}(\gamma )+\mathrm{f}(\gamma )\mathrm{f}(\alpha )$

(c) $\mathrm{f}(\alpha )\mathrm{f}(\beta )\mathrm{f}(\gamma )$

(d) $-\mathrm{f}(\alpha )\mathrm{f}(\beta )\mathrm{f}(\gamma )$

4. Let $a,b\mathrm{\&}c$ be such that $b(a+c)\ne 0$. If

$\left|\begin{array}{ccc}a& a+1& a-1\\ -b& b+1& b-1\\ c& c-1& c+1\end{array}\right|+\left|\begin{array}{ccc}a+1& b+1& c-1\\ a-1& b-1& c+1\\ (-1{)}^{n+2}a& (-1{)}^{n+1}a& (-1{)}^{n}c\end{array}\right|=0$, then the value of ’ $n$ ’ is

(a) zero

(b) any even integer

(c) any odd integer

(d) any integer

5. If $a,b,c$ are the sides of $\mathrm{△}\mathrm{ABC}$ such that $\left|\begin{array}{ccc}{\mathrm{a}}^2& {\mathrm{b}}^2& {\mathrm{c}}^2\\ (\mathrm{a}+1{)}^2& (\mathrm{b}+1{)}^2& (\mathrm{c}+1{)}^2\\ (\mathrm{a}-1{)}^2& (\mathrm{b}-1{)}^2& (\mathrm{c}-1{)}^2\end{array}\right|=0$ then

(a) $\mathrm{△}\mathrm{ABC}$ is non-isosceles right angled triangle

(b) $\mathrm{△}\mathrm{ABC}$ is an equilateral triangle

(c) $\mathrm{△}\mathrm{ABC}$ is an acute angled triangle with no two angles being equal

(d) $\mathrm{△}\mathrm{ABC}$ is an isosceles triangle

6. ${\mathrm{a}}_1\mathrm{x}+{\mathrm{b}}_1\mathrm{y}+{\mathrm{c}}_1\mathrm{z}+{\mathrm{d}}_1,=0,{\mathrm{a}}_2\mathrm{x}+{\mathrm{b}}_2\mathrm{y}+{\mathrm{c}}_2\mathrm{z}+{\mathrm{d}}_2=0,{\mathrm{a}}_3\mathrm{x}+{\mathrm{b}}_3\mathrm{y}+{\mathrm{c}}_3\mathrm{z}+{\mathrm{d}}_3=0$ which represent planes ${\mathrm{P}}_1$, ${\mathrm{P}}_2\mathrm{\&}{\mathrm{P}}_3$ respectively. Let

A=$\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]\text{ and }B=\left[\begin{array}{c}{a}_1{b}_1{c}_1{d}_1\\ a_2{b}_2{c}_2{d}_2\\ a_3{b}_3{c}_3{d}_3\end{array}\right]$ now match the entries form the following columns.

7. Let $\mathrm{M}$ be a $3\times 3$ matrix satisfying $\mathrm{M}\left[\begin{array}{l}010\end{array}\right]=\left[\begin{array}{l}-1\\ 2\\ 3\end{array}\right],\mathrm{M}\left[\begin{array}{l}1\\ -1\\ 0\end{array}\right]=\left[\begin{array}{l}1\\ 1\\ -1\end{array}\right]$ and $\mathrm{M}\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{l}0\\ 0\\ 12\end{array}\right]$, then the sum of the diagonal entries of $\mathrm{M}$ is

(a) 9

(b) 7

(c) 0

(d) none of these

8.* For all values of $\lambda$, the rank of the matrix $A=\left[\begin{array}{ccc}1& 4& 5\\ \lambda & 8& 8\lambda -6\\ 1+{\lambda }^2& 8\lambda +4& 2\lambda +21\end{array}\right]$

(a) for $\lambda =2,\rho (\mathrm{A})=1$

(b) $\lambda =-1,\rho (\mathrm{A})=2$

(c) for $\lambda \ne 2,-1,\rho (\mathrm{A})=3$

(d) none of these

9. Read the passage and answer the following questions.

If ${\mathrm{e}}^{\mathrm{A}}$ is defined as ${\mathrm{e}}^{\mathrm{A}}=\mathrm{I}+\mathrm{A}+{\displaystyle \frac{{\mathrm{A}}^2}{2!}}+{\displaystyle \frac{{\mathrm{A}}^3}{3!}}+\dots \dots \dots$

$={\displaystyle \frac{1}{2}}\left[\begin{array}{lll}f(x)& g(x)g(x)& f(x)\end{array}\right]$ where $A=\left[\begin{array}{lll}x& xx& x\end{array}\right]$ and $0<x<1$, then $I$ is an identity function.

i. $\int {\displaystyle \frac{g(x)}{f(x)}}dx$ is equal to

(a) $\log ({\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}})+\mathrm{C}$

(b) $\log ({\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}})+\mathrm{C}$

(c) $\log ({\mathrm{e}}^{2\mathrm{x}}-1)+\mathrm{C}$

(d) none of these

ii. $\int (g(x)+1)\sin xdx$ is equal to

(a) ${\displaystyle \frac{e^x}{2}}(\sin x-\cos \mathrm{x})+C$

(b) ${\displaystyle \frac{{\mathrm{e}}^{2\mathrm{x}}}{5}}(2\sin \mathrm{x}-\cos \mathrm{x})+\mathrm{C}$

(c) ${\displaystyle \frac{e^x}{5}}(\sin 2x-\cos 2x)+C$

(d) none of these

iii. $\int {\displaystyle \frac{f(x)}{\sqrt{g(x)}}}dx$ is equal to

(a) ${\displaystyle \frac{1}{2\sqrt{{\mathrm{e}}^{\mathrm{x}}-1}}}-{\mathrm{cosec}}^{-1}\left({\mathrm{e}}^{\mathrm{x}}\right)+\mathrm{C}$

(b) ${\displaystyle \frac{2}{2\sqrt{{\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}}}}}-{\sec }^{-1}\left({\mathrm{e}}^{\mathrm{x}}\right)+\mathrm{C}$

(c) ${\displaystyle \frac{1}{2\sqrt{{\mathrm{e}}^{2x}-1}}}+{\sec }^{-1}\left({\mathrm{e}}^{\mathrm{x}}\right)+\mathrm{C}$

(d) none of these

10. Consider three matrices $\mathrm{A}=\left[\begin{array}{ll}2& 1\\ 4& 1\end{array}\right],\mathrm{B}=\left[\begin{array}{ll}3& 4\\ 2& 3\end{array}\right]$ and $\mathrm{C}=\left[\begin{array}{cc}3& -4\\ -2& 3\end{array}\right]$. Then the value of the sum $\mathrm{tr}(\mathrm{A})+\mathrm{tr}\left({\displaystyle \frac{\mathrm{ABC}}{2}}\right)+\mathrm{tr}\left({\displaystyle \frac{\mathrm{A}(\mathrm{BC}{)}^3}{4}}\right)+\mathrm{tr}\left({\displaystyle \frac{\mathrm{A}(\mathrm{BC}{)}^3}{8}}\right)+\dots \dots$. is

(a) 6

(b) 9

(c) 12

(d) none of these

11. If $\mathrm{A}=\left[\begin{array}{lll}2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\end{array}\right]$, then ${\mathrm{A}}^3-7{\mathrm{A}}^2+10\mathrm{A}=$

(a) $5\mathrm{I}+\mathrm{A}$

(b) $5\mathrm{I}-\mathrm{A}$

(c) $\mathrm{A}-5\mathrm{I}$

(d) $6\mathrm{I}$

12. If $y=\exp (({\sin }^{2}x+{\sin }^{4}x+{\sin }^{6}x+\dots ..){\log }_{e}2)$ satisfies $x^2-17x+16=0$, where $0<x<{\displaystyle \frac{\pi }{2}}$ and if ${\displaystyle \frac{2\sin 2x}{1+{\cos }^{2}x}}=\alpha ,{\displaystyle \frac{2\sin x}{\sin x+\cos x}}=\beta ,\sum _{n=1}^{\mathrm{\infty }}(\cot x{)}^n=\gamma ,\sum _{n=1}^{\mathrm{\infty }}(\cot x{)}^{2n}=\delta$ then the sum of all the elements of the matrix $\left[\begin{array}{ll}3\alpha & 3\beta \\ 4\gamma & 9\delta \end{array}\right]$ is

(a) a perfect square

(c) 16

(b) not a perfect square

(d) 15

13. If $\mathrm{x},\mathrm{y},\mathrm{z},\mathrm{w}\in \mathrm{R}$ satisfy the following equations.

$x+y+z+w=1;x+2y+4z+8w=16,x+3y+9z+27w=81$ and $x+4y+16z+64w=256$, then the pairs

which have H.C.F as 2 is

(a) $|\mathrm{w}|,|\mathrm{z}|$

(b) $|w|,|y|$

(c) $|y|,|x|$

(d) $|z|,|x|$

14. If $(\alpha \beta )(\delta \beta )=\gamma \gamma \gamma$ such that $\alpha ,\beta ,\delta ,\gamma$ represent a number from 1 to 9 and are different digits and $\alpha \beta ,\delta \beta$ are two digit numbers and $\gamma \gamma \gamma$ is three digit number and the trace of the matrix $\mathrm{A}=$ $\left[\begin{array}{llll}\alpha & 1& 2& 0\\ 0& \beta & 1& 1\\ 0& 0& \gamma & 3\\ 1& 1& 0& \delta \end{array}\right]$ is a, then ${\displaystyle \frac{a}{7}}=$

(a) 21

(b) 7

(c) 3

(d) none of these

15. Let $\mathrm{A}=\left[\begin{array}{cc}1& 3/2\\ 1& 2\end{array}\right],\mathrm{B}=\left[\begin{array}{cc}4& -3\\ -2& 2\end{array}\right]$ and ${\mathrm{C}}_{\mathrm{r}}=\left[\begin{array}{cc}\mathrm{r}\cdot 3^{\mathrm{r}}& 2^{\mathrm{r}}\\ 0& (\mathrm{r}-1)3^{\mathrm{r}}\end{array}\right]$ be given three matrices, then the value of $\sum _{\mathrm{r}=1}^{50}\mathrm{tr}((\mathrm{AB}{)}^{\mathrm{r}}{\mathrm{C}}_{\mathrm{r}})=\mathrm{a}{.3}^{\mathrm{b}}+3$ then ${\displaystyle \frac{\mathrm{a}+\mathrm{b}}{25}}=$

(a) 4

(b) 16

(c) 49

(d) none of these