2000

1) Let $V$ be a vector space over $\mathrm{R}$ and Let $\mathrm{T}=\{(\mathrm{x},\mathrm{y})\mid \mathrm{x},\mathrm{y},\in \mathrm{V}\}$Define addition in T componentwisc and scalar multiplication by a complex number $\alpha +\mathrm{i}\beta$ by $(\alpha +i\beta )(x,y)=(\alpha x-\beta y,\beta x+\alpha y)$ $\mathrm{\forall }\alpha ,\beta \in R$. Show that $\mathrm{T}$ is a vector space over $C$.

[10M]

2) Show that if $\lambda$ is a characteristic root of a non-singular matrix $A,$ then ${\lambda }^{-1}$ is a characteristic root of $A^{-1}$.

[10M]

3) Use the Mean value theorem to prove ${\displaystyle \frac{2}{7}}<\log 1.4<{\displaystyle \frac{2}{5}}$.

[10M]

4) Show that $\iint x^{2l-1}{y}^{2m-1}dxdy={\displaystyle \frac{1}{4}}{r}^{2(l+m)}{\displaystyle \frac{\mathrm{\Gamma }l\mathrm{\Gamma }m}{\mathrm{\Gamma }(l+m+1)}}$ for all positive values of $x$ and $y$ lying inside the circle $x^2+y^2=r^2$.

[10M]

5) Find the equations of the planes bisecting the angles between the planes $2x-y-2z-3=0$ and $3x+4y+1=0$ and specify the one which bisects the açute angle.

[10M]

6) Find the equation to the common conjugate diameters of the conics $x^2+4xy+6y^2=1$ and $2x^2+6xy+9y^2=1$.

[10M]

7) Prove that a real symmetnc matrix A is positive definite if and only $\mathrm{A}={\mathrm{B}\mathrm{B}}^t$ for some non-singular matrix $\mathrm{B}$. Show also that $A=\left(\begin{array}{lll}1& 2& 3\\ 2& 5& 7\\ 3& 7& 11\end{array}\right)$ is positive definite and find the matrix $B$ such that $A=B{B}^t$. Here $B^t$ stands for the transpose of $\mathrm{B}$.

[10M]

8) Prove that a system $\mathrm{A}\mathrm{X}=\mathrm{B}$ of $\mathrm{n}$ non-homogeneous equations in $\mathrm{n}$ unknowns has a unique solution provided the coefficient matrix is non singular.

[10M]

9) Prove that two similar matrices have the same characteristic roots. Is its converse true? Justify your claim.

[10M]

2.d) Reduce the equation $x^2+y^2+z^2-2xy-2yz+2zx+x-y-2z+6=0$ into canonical form and determine the nature of the quadratic.

[10M]

1999

1) Let $V$ be the vector space of functions from $R$ to $R$ (the real numbers).Show that $f,g,h$ in $V$ are linearly independent were $f(t)=e^{2t}$, $g(t)=t^{2}andh(t)=t$.

[10M]

2) If the matrix of a linear transformation $T$ on $V_2(R)$ with repect to the basis $B=\{(1,0),(0,1)\}$ is $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then what is the matrix of $T$ with respect to the ordered basis $B_1=\{(1,1),(1,-1)\}$?

[10M]

3) Diagonalize the matrix $A=\left[\begin{array}{ccc}4& 2& 2\\ 2& 4& 2\\ 2& 2& 4\end{array}\right]$

[10M]

4) Test for congruency of the matrices $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}0& i\\ -i& 0\end{array}\right]$.Prove that $A^{2m}=B^{2m}=I$ where $m$ and $n$ are positive integers.

[10M]

5) If $A$ is a skew symmetric matrix of order $n$, prove that $(I-A)(I+A{)}^{-1}$ is orthogonal.

[10M]

6) Test for the positive definiteness of the quadratic form $2x^2+y^2+2z^2+2xy-2zx$

[10M]

1998

1) Given two linearly independent vectors (1,0,1,0) and (0,-1,1,0) of $R^4$, find a basis $R^4$ which includes these two vectors.

[10M]

2) If $V$ is a finite dimensional vector space over $R$ and if f and g are two linear transformation from $V$ to $R$ such that $f(v)=0$ implies $g(v)=0$ then prove that $g=\lambda$f for soome $\lambda$ in $R$.

[10M]

3) Let T:$R^3\to R^3$ be defined by $T(x_1,x_2,x_3)=(x_2,x_3-c{x}_1-b{x}_2-a{x}_3)$, where a,b,c are fixed real numbers.Show that T is a linear transformation of $R^3$ and that $A^3+a{A}^2+bA+cI=0$ ,where $A$ is the matrix of $T$ with respect to standard basis of $R^3$

[10M]

4) If $A$ and $B$ are two matrices of order $2\times 2$ such that $A$ is skew Hermitian and $AB=B$, then show that $B=0$.

[10M]

5) If $T$ is a complex matrix of order $2\times 2$ such that T=tr $T^2=0$, then show that $T^2=0$.

[10M]

6) Prove that a necessary and sufficient condition for a $n\times n$ real matrix $A$ to be similar to a diagonal matrix is that the set of characteristic vectors of $A$ includes a set of $n$ linearly independent vectors.

[10M]

7) Let $A$ be an $m\times n$ matrix. Then show that the sum of the real and nullity of $A$ is $n$.

[10M]

8) Find all real $2\times 2$ matrices $A$ whose characteristic roots are real and which satisfy $A{A}^{\prime }=I$.

[10M]

9) Reduce to diagonal martrix by rational congruent transformation the symmetric matrix $A=\left[\begin{array}{ccc}1& 2& -1\\ 2& 0& 3\\ -1& 3& 1\end{array}\right]$

[10M]

1997

1) Let $\mathrm{V}$ be the vector space of polynomials over $\mathrm{R}$. Find a basis and dimension of the subspace $\mathrm{W}$ of $\mathrm{V}$ spanned by the polynomials

${\mathrm{v}}_1={\mathrm{t}}^3-2{\mathrm{t}}^2+4\mathrm{t}+1$, ${\mathrm{v}}_2=2{\mathrm{t}}^3-3{\mathrm{t}}^2+9\mathrm{t}-1$, ${\mathrm{v}}_3={\mathrm{t}}^3\mathrm{t}-5$, ${\mathrm{v}}_4=2t^3-5{\mathrm{t}}^2+7\mathrm{t}+5$.

[10M]

2) Verify that the transformation defined by $\mathrm{T}({\mathrm{x}}_1,{\mathrm{x}}_2)=({\mathrm{x}}_1+{\mathrm{x}}_2,{\mathrm{x}}_1-{\mathrm{x}}_2,{\mathrm{x}}_2)$ is a linear transformation from ${\mathrm{R}}^2$ into ${\mathrm{R}}^3$. Find its range, null space and nullity.

[10M]

3) Let $\mathrm{V}$ be the vector space of $2\times 2$ matrices over $\mathrm{R}$. Determine whether the matrices $\mathrm{A},\mathrm{B},\mathrm{C}$ $\in \mathrm{V}$ are dependent where $A=\left[\begin{array}{ll}1& 2\\ 3& 1\end{array}\right],B=\left[\begin{array}{cc}3& -1\\ 2& 2\end{array}\right],C=\left[\begin{array}{cc}1& -5\\ -4& 0\end{array}\right]$

[10M]

4) Let a square matrix A of order $n$ be such that each of its diagonal elements is $\mu$ and each of its off diagonal elements is 1 . If $B=\lambda A$ is orthogonal, determine the values of $\lambda$ and $\mu$.

[10M]

5) Show that $A=\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& 0\\ 2& 2& 3\end{array}\right]$ is diagoalisable over $\mathrm{R}$ and find a matrix $\mathrm{P}$ such that ${\mathrm{P}}^{-1}\mathrm{A}\mathrm{P}$ is diagonal. Hence determine $A^{25}$.

[10M]

6) Let $A=\left[a_j\right]$ be a square matrix of order $n$ such that $\left[a_{ij}\right]\le M\mathrm{\forall }i,j=1,2,\dots ,n$. Let $\lambda$ be an eigen-value of $A$, Show that $|\lambda |\le \mathrm{n}\mathrm{M}$.

[10M]

7) Define a positive definite matrix. Show that a positive definite matrix is always non-singular. Prove that its converse does not hold.

[10M]

8) Find the characteristic roots and their corresponding vectors for the matrix $\left[\begin{array}{ccc}6& -2& 2\\ -2& 3& -1\\ 2& -1& 3\end{array}\right]$

[10M]

9) Find an invertible matrix $\mathrm{P}$ which reduces $Q(x,y,z)=2xy+2yz+2x$ to its canonical form.

[10M]

1996

1) ${\mathrm{R}}^4,$ let ${\mathrm{W}}_1$ be the space generated by (1,1,0,-1),(2,6,0) and (-2,-3,-3,1) and let ${\mathrm{W}}_2$ be the space generated by $(-1,-2,-2,2),(4;,6,4,-6)$ and $(1,3,4,-3)$ . Find a basis for the space $W_1+W_2$

[12M]

2) Let $\mathrm{V}$ be a finite dimensional vector space and $\mathrm{v}\in \mathrm{V},\mathrm{v}\ne 0.$ Show that there exists a linear functional $f$ on $\mathrm{V}$ such that $f(v)\ne 0.$

[12M]

3) Let $\mathrm{V}={\mathrm{R}}^3$ and ${\mathrm{v}}_1,{\mathrm{v}}_2,{\mathrm{v}}_3$ be a basis of ${\mathrm{R}}^3$. Let $\mathrm{T}:\mathrm{V}\to \mathrm{V}$ be a linear transformation such that $\mathrm{T}\left({\mathrm{v}}_1\right)={\mathrm{v}}_1+{\mathrm{v}}_2+{\mathrm{v}}_3,1\le \mathrm{i}\le 3.$ By writing the matrix of $\mathrm{T}$ with respect to another basis, show that the matrix $\left[\begin{array}{lll}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$ is similar to $\left[\begin{array}{lll}3& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

[12M]

4) Let $\mathrm{V}={\mathrm{R}}^3$ and $\mathrm{T}:\mathrm{V}\to \mathrm{V}$ be the linear map defined by $\mathrm{T}(\mathrm{x},\mathrm{y},z)=(\mathrm{x}+2,-2\mathrm{x}+\mathrm{y},-\mathrm{x}+2\mathrm{y}+\mathrm{z})$ What is the matrix of T with respect to the basis (1,0,1),(-1,1,1) and (0,1,1)$?$ Using this matrix, write down the matrix of T with respect to the basis (0,1,2),(-1,1,1) and (0,1,1).

[15M]

5) Let $\mathrm{V}$ and $\mathrm{W}$ be finite dimensional vector spaces such that $\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{V}\ge \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{W}$. Show that there is always a linear map of $\mathrm{V}$ onto $\mathrm{W}$.

[15M]

6) Solve

by using Cramer’s rule.

[15M]

7) Find the inverse of the matrix $\left[\begin{array}{llll}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\end{array}\right]$ by computing its characteristic polynomial.

[15M]

8) Let $A$ and $B$ be $n\times n$ matrices such that $AB=BA$. Show that $A$ and $B$ have a common characteristic vector.

[15M]

9) Reduce to canonical form the orthogonal matrix

[15M]

1995

1) Let $T$ be the linear operator in ${\mathrm{R}}^3$ defined by $\mathrm{T}({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3)=(3{\mathrm{x}}_1+{\mathrm{x}}_3,-2{\mathrm{x}}_1+{\mathrm{x}}_{2,}-{\mathrm{x}}_1+2{\mathrm{x}}_2+4{\mathrm{x}}_3)$ What is the matrix of $T$ in the standard ordered basis for $R^3?$ What is a basis of range space of T and a basis of null space of T?

[20M]

2) Let $A$ be a square matrix of order $\mathrm{n}$. Prove that $\mathrm{A}\mathrm{X}=\mathrm{b}$ bas a solution if and only if be ${\mathrm{R}}^{\mathrm{\prime }\mathrm{\prime }}$ is orthogonal to all solutions Y of the svstem $A^{\mathrm{\top }}Y=0$.

[20M]

3) Define a similar matrix. Prove that the characteristic equation of two similar matrices is the same. Let 1,2,3 be the eigen-values of a matrix. Write down such a matrix. Is such a matrix unique?

[20M]

4) Show that $A=\left[\begin{array}{ccc}5& -6& -6\\ -1& 4& 2\\ 3& -6& -4\end{array}\right]$ is diagonalizable and hence determine $A^3$.

[20M]

5) Let $A$ and $B$ be matrices of order $n$ Prove that if $(I-AB)$ is invertible, then $(I-BA)$ is also invertible and $(1-BA{)}^{-1}=1+B(1-AB{)}^{-1}A$

[20M]

6) If $a$ and $b$ are complex numbers such that $|\mathrm{b}|=\mathrm{I}$ and $\mathrm{H}$ is a Hermitian matrix, show that the eigen-values of $$aI+$bH$ lie on a straight line in the complex plane.

[20M]

7) Let $A$ be a symmefric matrix. Show that $A$ is positive definite if and only if its eigen-values are all positive

[20M]

8) Let $A$ and $B$ be square matrices of order $n$. Show that $AB$ -BA can never be equal to unit matrix.

[20M]

9) Let $\mathrm{A}=\left[a_{ij}\right];\mathrm{i},\mathrm{j}=1,2,\dots ,\mathrm{n}$ and $\left|a_n\right|>\sum _{j\ne i}\left|a_{ij}\right|j=1,2,\dots ,n$ for every $i=1,\mathrm{2...},n$. Show that A is non-singualr matrix. Hence or otherwise prove that the eigen-values of A lie in the discs $|\lambda -a_n|\le \sum _{j\ne i}\left|a_{ij}\right|j=1,2,\dots ,n$ in the complex plane.

[20M]