[AKU- Year 2020]
Q1. If $A=\begin{bmatrix}2 &0&0\\0&2&0\\0&0&2\end{bmatrix}~ and~ B=\begin{bmatrix}1&2&3\\0&1&3\\0&0&2\end{bmatrix}$
Then the determinant of $AB$ has the value
a) 4 b) 8 c) 16 d) 32
Q2. The sum and product of the eigenvalues of
$\begin{bmatrix}2&2&1\\1&3&1\\1&2&2\end{bmatrix}$ are
a) 7 and 5 b) 5 and 7 c) 12 and 3 d) 3 and 12
[AKU- Year 2018]
Q3. If the nullity of the matrix $\begin{bmatrix}k&1&2\\1&-1&-2\\1&1&4 \end{bmatrix}$ is 1. Then the value of $k$ is
a) -1 b) 0 c) 1 d) 2
Q4. Let $A=\begin{bmatrix}3&0&0\\0&6&2\\0&2&6\end{bmatrix}$ and let $\lambda_1 \geq \lambda_2 \geq \lambda_3$ be the eigenvalues of $A$. Then the triple $(\lambda_1, \lambda_2, \lambda_3)$equals
a) (9, 4, 2) b) (8,4,3) c) (9,3,3) d)(7,5,3)
Q5. If $A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\end{bmatrix}$ then $A^{50}$ is
a) $\begin{bmatrix}1&0&0\\50&0&0\\50&0&1\end{bmatrix}$ b) $\begin{bmatrix}1&0&0\\48&0&0\\48&0&1\end{bmatrix}$ c) $\begin{bmatrix}1&0&0\\25&1&0\\25&0&1\end{bmatrix}$ d)$\begin{bmatrix}1&0&0\\24&1&0\\24&0&1\end{bmatrix}$
[AKU- Year 2019]
Q6. If the eigen values of the given matrix
$\begin{bmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{bmatrix}$
is 3. then the eigen value of $adj (A)$ will be
a) $-\frac{1}{3}$ b) $-\frac{1}{5}$ c)$-\frac{1}{15}$ d) -3
Q7. Write down the quadratic forms corresponding to the given matrix
$\begin{bmatrix}2&4&5\\4&3&1\\5&1&1\end{bmatrix}$
Subjective Questions:
AKU- YEAR 2020
Q1. Determine the value of $p$ such that the rank of
$\begin{bmatrix}1&1&-1&0\\4&4&-3&1\\p&2&2&2\\9&9&p&3\end{bmatrix}$ is 3
Solution: Click here for solution.
Q2. Use Gauss-Jordan method to find the inverse of the matrix
$\begin{bmatrix}2&3&4\\4&3&1\\1&2&4\end{bmatrix}$
Solution: Click here for solution.
Q3. Find the non-singular matrices $P$ and $Q$ such that
$\begin{bmatrix}1&2&3&4\\2&1&4&3\\3&0&5&-10\end{bmatrix}$
is reduced to normal form. Also find its rank.
Solution: Click here for solution.
Q4. Solve the given equations by Cramer's rule
$x+y+z=4$
$x-y+z=0$
$2x+y+z=5$
Solution: Click here for solution.
Q5. Verify Cayley-Hamilton theorem for the matrix
$\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}$ and find the inverse.
Solution: Click here for solution.
Q6. Find the eigen vectors of the matrix
$\begin{bmatrix}6&-2&2\\-2&3&-1\\2&-1&3\end{bmatrix}$
Hence deduce
$6x^2+3y^2+3z^2-2yz+4zx-4xy$
to a 'sum of squares'. Also write nature of the matrix.
Solution: Click here for solution.
Q7. Find the eigen values and eigen vectors of the matrix
$\begin{bmatrix}-2&2&-3\\2&1&-6\\-1&-2&0\end{bmatrix}$
Solution: Click here for solution.
Q8. For what values of $k$, the equations
$x+y+z=1,~2x+y+4z=k,~4x+y+10z=k^2$
Have solution? Solve them completely in each case.
Solution: Click here for solution.
Q9. Reduce the matrix
$\begin{bmatrix}-1&2&-2\\1&2&1\\-1&-1&0\end{bmatrix}$ to the diagonal form.
Solution: Click here for solution.
Q10. Find the rank of the matrix
$\begin{bmatrix}2&3&-1&-1\\1&-1&-2&-4\\3&1&3&-2\\6&3&0&-7\end{bmatrix}$
Solution: Click here for solution.
Q11. Find the characteristic equation of the matrix
$A=\begin{bmatrix}2&1&1\\0&1&0\\1&1&2\end{bmatrix}$
and hence find the matrix represented by
$A^8-5A^7+7A^6-3A^5+A^4-5A^3+8A^2-2A+I$
Solution: Click here for solution.
Q12. State and prove Cayley-Hamilton theorem.
Solution: Click here for solution.
Q13. Reduce the quadratic form $3x^2+5y^2+3z^2-2xy-2yz+2zx$ to canonical forms.
Solution: Click here for solution.
AKU- YEAR 2018
Q14. Let $T:R^3\rightarrow R^4$ be a linear transformation defined by
$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}x+y\\z+y\\x+z\\x+y+z\end{bmatrix}$
Find the matrix representation of $T$ with respect to the ordered basis
$x=\lbrace \begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}\rbrace$ in $R^3$ and $y=\lbrace \begin{bmatrix}0\\1\\1\\1\end{bmatrix},\begin{bmatrix}1\\0\\1\\1\end{bmatrix},\begin{bmatrix}1\\1\\0\\1\end{bmatrix},\begin{bmatrix}1\\1\\1\\0\end{bmatrix}\rbrace $ in $R^4$
Solution: Click here for solution.
