PYQs

1) Show that similar matrices have the same characteristic polynomial.

[2017, 10M]

2) If $A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$ then find $A^{14}+3A-2I$.

[2016, 4M]

3) If $\lambda$ is a characteristic root of a non-singular matrix $A$, then prove that ${\displaystyle \frac{|A|}{\lambda }}$ is a characteristic root of $\mathrm{Adj}A$.

[2012, 8M]

4) Find the characteristic polynomial of the matrix $A=\left[\begin{array}{cc}1& 1\\ -1& 3\end{array}\right]$. Hence, find $A^{-1}$ and $A^6$.

[2004, 15M]

5) A square matrix $A$ is non-singular if and only if the constant term in its characteristic polynomial is different from zero.

[2002, 12M]

6) If $\lambda$ is a characteristic root of a non-singular matrix $A$ then prove that ${\displaystyle \frac{|A|}{\lambda }}$ is a characteristic root of $Adj(A)$.

[2001, 12M]

1) Show that if $A$ and $B$ are similar $n\times n$ matrices, then they have the same eigenvalues.

[2018, 12M]

2) Prove that distance non-zero eigenvectors of a matrix are linearly independent.

[2017, 10M]

3) If $A=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]$, then find the Eigen values and Eigenvectors of $A$.

[2016, 6M]

4) Find the Eigen values and Eigen vectors of the matrix $\left[\begin{array}{ccc}1& 1& 3\\ 1& 5& 1\\ 3& 1& 1\end{array}\right]$.

[2015, 12M]

5) Let $A=\left[\begin{array}{ccc}-2& 2& -3\\ 2& 1& -6\\ -1& -2& 0\end{array}\right]$. Find the Eigen values of $A$ and the corresponding Eigen vectors.

[2014, 8M]

6) Let $A$ be a square matrix and $A^{\ast }$ be its adjoint, show that the Eigen values of matrices $A{A}^{\ast }$ and $A^{\ast }A$ are real. Further show that $\mathrm{trace}\left(A{A}^{\ast }\right)=\mathrm{trace}\left(A^{\ast }A\right)$.

[2013, 10M]

7) Let $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& {\omega }^2& \omega \\ 1& \omega & {\omega }^2\end{array}\right]$, where $\omega (\ne 1)$ is a cube root of unity. If ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ denote the Eigen values of $A^2$, show that $|\lambda {|}_1+|{\lambda }_2|+|{\lambda }_3|\le 9$.

[2013, 8M]

8) Let ${\lambda }_1$, ${\lambda }_2$, $\dots \dots {\lambda }_n$ be the Eigen values of a $n\times n$ square matrix $A$ with corresponding Eigen vectors $X_1,X_2,\dots .X_n$. If $B$ is a matrix similar to $A$, show that the Eigen values of $B$ are same as that of $A$. Also, find the relation between the Eigen vectors of $\mathrm{B}$ and Eigen vectors of $\mathrm{A}$.

[2011, 10M]

9) Let $A=\left[\begin{array}{ccc}2& -2& 2\\ 1& 1& 1\\ 1& 3& -1\end{array}\right]$ and $C$ be a non-singular matrix of order $3\times 3$. Find the Eigen values of the matrix $B^3$ where $B=C^{-1}AC$

[2011, 10M]

10) If ${\lambda }_1$, ${\lambda }_2$, $\dots {\dots }_3$ are the Eigen values of the matrix $\mathrm{A}=\left[\begin{array}{ccc}26& -2& 2\\ 2& 21& 4\\ 44& 2& 28\end{array}\right]$, show that $\sqrt{{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2}\le \sqrt{1949}$

[2010, 12M]

11) Let $A$ and $B$ be $n\times n$ matrices over reals. Show that $I-BA$ is invertible if $I-AB$ is invertible. Deduce that $AB$ and $BA$ have the same Eigen values.

[2010, 20M]

12) Prove that the Eigen vectors corresponding to distinct Eigen values of a square matrix are linearly independent.

[2003, 15M]

13) Let $A$ be a real $3\times 3$ symmetric matrix with Eigen values 0, 0 and 5. If the corresponding Eigen-vectors are (2,0,1), (2,1,1) and (1,0,-2), then find the matrix $A$.

[2002, 15M]

1) State the Cayley-Hamilton theorem. Use this theorem to find $A^{100}$, where

[15M]

2) If matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ then find $A^{30}$.

[2015, 12M]

3) Verify the Cayley-Hamilton theorem for the matrix

Using this, show that A is non-sinugular and find A^{-1}.

[2011, 10M]

4) Verify Cayley-Hamilton theorem for the matrix $A=\left[\begin{array}{cc}1& 4\\ 2& 3\end{array}\right]$ and hence find its inverse. Also, find the matrix represented by $A^5-4A^4-7A^3+11A^2-A-10I$.

[2014, 10M]

5) State Cayley-Hamilton theorem and using it, find the inverse of $\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$.

[2006, 12M]

6) If $A=\left[\begin{array}{ccc}2& 1& 1\\ 0& 1& 0\\ 1& 1& 2\end{array}\right]$, then find the matrix represented by $2A^{10-}10A^9+14A^8-6A^7-3A^6+15A^5-21A^4+9A^3+A-1$.

[2003, 12M]

7) Use Cayley-Hamilton theorem to find the inverse of the following matrix: $\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$.

[2002, 15M]

8) If $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$, show that for every integer $n\ge 3$, $A^n=A^{n-2}+A^2-I$. Hence, determine $A^{50}$.

[2001, 15M]