AKU Previous year questions from Matrices:

 [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.