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 $A X=$ $\lambda X, \lambda$ is called an eigen value of $A \& X$, 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 $A=\sum\limits _{\mathrm{I}=1}^{\mathrm{n}} \mathrm{a} _{\mathrm{ii}}$
• 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 $\hspace {2 cm}\mathrm{a} _{11} \mathrm{x}+\mathrm{a} _{12} \mathrm{y}+\mathrm{a} _{13} \mathrm{z}=\mathrm{b} _{1}$

$\hspace {3 cm}\begin{aligned} & a _{21} x+a _{22} y+a _{23} z=b _{2} \\ & a _{31} x+a _{32} y+a _{33} z=b _{3} \end{aligned} \quad \text { is given }$

by $X=A^{-1} B \quad$ 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 $|\mathrm{A}| \neq 0$, the system is consistent and have unique solution.

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

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

If (adj A) $B \neq 0$ 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 $1^{\text {st }}, 2^{\text {nd }}, 3^{\text {rd }}, \ldots$ 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

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

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

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

$\left[\begin{array}{lll}a _{11} & a _{12} & a _{13} \\ a _{21} & a _{22} & a _{23} \\ a _{31} & a _{32} & a _{33}\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left|\begin{array}{l}b _{1} \\ b _{2} \\ b _{2}\end{array}\right|, A X=B$

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

$[A B]=\left[\begin{array}{llll}a _{11} & a _{12} & a _{13} & b _{1} \\ a _{21} & a _{22} & a _{23} & b _{2} \\ a _{31} & a _{32} & a _{33} & b _{3}\end{array}\right]$

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 $\mathrm{r} \neq \mathrm{s}$, 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 $\mathrm{P}^{\prime}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be the reflection of $\mathrm{P}(\mathrm{x}, \mathrm{y})$ on $\mathrm{x}$-axis

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

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

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

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

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

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

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

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

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

iii. Reflection through the origin

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

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

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

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

iv. Reflection in the line $y=x$

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

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

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

$\therefore \quad\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$ 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 $\mathrm{y}=\mathrm{mx})$

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

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

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

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

$\mathrm{y}=\mathrm{x} \tan \alpha . \quad($ 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.

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

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

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

$\therefore \quad\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ 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 $\mathrm{A}^{-1}=\dfrac{1}{6}\left(\mathrm{~A}^{2}-\mathrm{cA}+\mathrm{dI}\right)$, 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 $\mathrm{A}=\left[\begin{array}{cc}-1 & \dfrac{3}{2} \\ \dfrac{-1}{2} & \dfrac{1}{2}\end{array}\right]$, then $\mathrm{A}^{3}=$

(a) $\dfrac{\mathrm{A}}{4}$

(b) $\dfrac{\mathrm{A}}{8}$

(c) $\dfrac{\mathrm{I}}{4}$

(d) $\dfrac{\mathrm{I}}{8}$

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

(a) 0

(b) 1

(c) 2

(d) Infinite

5. If the system of equations $x-k y-z=0, k x-y-z=0$ 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 $x+2 a y+a z=0, x+3 b y+b z=0$ and $x+4 c y+c z=0$ 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

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

$x+\alpha y+z=\alpha-1 \quad$ 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 $x+y-1=0,2 x-y-c=0$ and $-b x+3 b y-c=0$ is consistent, then the possible real values of $b$ are

(a) $\mathrm{b} \in\left(-3, \dfrac{3}{4}\right)$

(b) $\mathrm{b} \in\left(-\dfrac{3}{2}, 4\right)$

(c) $\mathrm{b} \in\left(-\dfrac{3}{4}, 3\right)$

(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 $a x+\alpha y+z=0, b x+\beta y+z=0, a x+\gamma y+z=0$ have only zero solution. Then which of the following is not true?

(a) $\mathrm{r} _{1} \neq 1$

(b) $\mathrm{r} _{2} \neq 1$

(c) $r _{1} \neq r _{2}$

(d) none of these

3. If $f(x)=a+b x+c x^{2}$ 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 $a, b \& c$ be such that $b(a+c) \neq 0$. 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 $\triangle \mathrm{ABC}$ 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) $\triangle \mathrm{ABC}$ is non-isosceles right angled triangle

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

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

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

6. $\mathrm{a} _{1} \mathrm{x}+\mathrm{b} _{1} \mathrm{y}+\mathrm{c} _{1} \mathrm{z}+\mathrm{d} _{1},=0, \mathrm{a} _{2} \mathrm{x}+\mathrm{b} _{2} \mathrm{y}+\mathrm{c} _{2} \mathrm{z}+\mathrm{d} _{2}=0, \quad \mathrm{a} _{3} \mathrm{x}+\mathrm{b} _{3} \mathrm{y}+\mathrm{c} _{3} \mathrm{z}+\mathrm{d} _{3}=0$ which represent planes $\mathrm{P} _{1}$, $\mathrm{P} _{2} \& \mathrm{P} _{3}$ 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 $\mathrm{M}\left[\begin{array}{l}0 \ 1 \ 0\end{array}\right]=\left[\begin{array}{l}-1 \\ 2 \\ 3\end{array}\right], \mathrm{M}\left[\begin{array}{l}1 \\ -1 \\ 0\end{array}\right]=\left[\begin{array}{l}1 \\ 1 \\ -1\end{array}\right]$ 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 $\lambda \neq 2,-1, \rho(\mathrm{A})=3$

(d) none of these

9. Read the passage and answer the following questions.

If $\mathrm{e}^{\mathrm{A}}$ is defined as $\mathrm{e}^{\mathrm{A}}=\mathrm{I}+\mathrm{A}+\dfrac{\mathrm{A}^{2}}{2 !}+\dfrac{\mathrm{A}^{3}}{3 !}+\ldots \ldots \ldots$

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

i. $\quad \int \dfrac{g(x)}{f(x)} d x$ is equal to

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

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

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

(d) none of these

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

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

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

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

(d) none of these

iii. $\quad \int \dfrac{f(x)}{\sqrt{g(x)}} d x$ is equal to

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

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

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

(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 $\operatorname{tr}(\mathrm{A})+\operatorname{tr}\left(\dfrac{\mathrm{ABC}}{2}\right)+\operatorname{tr}\left(\dfrac{\mathrm{A}(\mathrm{BC})^{3}}{4}\right)+\operatorname{tr}\left(\dfrac{\mathrm{A}(\mathrm{BC})^{3}}{8}\right)+\ldots \ldots$. 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 $y=\exp \left(\left(\sin ^{2} x+\sin ^{4} x+\sin ^{6} x+\ldots ..\right) \log _{e} 2\right)$ satisfies $x^{2}-17 x+16=0$, where $0<x<\dfrac{\pi}{2}$ and if $\dfrac{2 \sin 2 x}{1+\cos ^{2} x}=\alpha, \dfrac{2 \sin x}{\sin x+\cos x}=\beta, \sum\limits _{n=1}^{\infty}(\cot x)^{n}=\gamma, \sum\limits _{n=1}^{\infty}(\cot x)^{2 n}=\delta$ 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.

$x+y+z+w=1 ; x+2 y+4 z+8 w=16, x+3 y+9 z+27 w=81$ and $x+4 y+16 z+64 w=256$, 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 $\dfrac{a}{7}=$

(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 $\mathrm{C} _{\mathrm{r}}=\left[\begin{array}{cc}\mathrm{r} \cdot 3^{\mathrm{r}} & 2^{\mathrm{r}} \\ 0 & (\mathrm{r}-1) \mathrm{3}^{\mathrm{r}}\end{array}\right]$ be given three matrices, then the value of $\sum\limits _{\mathrm{r}=1}^{50} \operatorname{tr}\left((\mathrm{AB})^{\mathrm{r}} \mathrm{C} _{\mathrm{r}}\right)=\mathrm{a} .3^{\mathrm{b}}+3$ then $\dfrac{\mathrm{a}+\mathrm{b}}{25}=$

(a) 4

(b) 16

(c) 49

(d) none of these