In this blog, we are going to discuss some frequently asked NET UGC questions about matrices.
These questions and answers will boost your study
Question 1
If \[A=\begin{bmatrix} 2&-2 &1 \\ 5 &-5 &1 \\ 1/5 &-3/5 &1/5 \end{bmatrix}\], then the sum of all elements of\[A^{200}\] is
(a)1
(b)2
(c)200
(d)3
Solution:
(d) 3
Here sum of the elemets of each row of A is 1
So the sum of elements in each row of $A^{200}$ is$1^200$
Then the sum of all elements of\[A^{200}\]is
\[1^{200}+1^{200}+1^{200}=3\]
Question 2
If \[A=\begin{bmatrix} 1 &2 &3 \\ 4 &5 &6 \\ 7 &8 &9 \\ 3 &1 &5 \end{bmatrix}\]
Then the number of submatrices of order1*2 is
(a) 4
(b) 12
(c) 6
(d) 7
Solution:
(b) 12
Number of sub matrices is \[\binom{m}{i}\times\binom{n}{j}\]
\[=\binom{4}{1}\times\binom{3}{2} =4\times3 =12\]
Question 3
Let A and B are two square matrices of same order such that AB=BA=0
then \[(A+B)^n=(A-B)^n\]if
(a) n= odd
(b) n= even
(c) n= odd or even
(d) none of these
Solution:
(a) n= even
\[(A+B)^n=(A-B)^n\]
\[A^n+B^n+(AB)+...=A^n+(-B)^n+...\]
\[A^n+B^n=A^n+(-B)^n\]
if n is even
Question 4
If A is real square matrix of odd order such that\[A+A^T=0\]
then
(a) A is singular
(b) A is non singular
(c) \[A+A^T\text{ is non singular}\]
(d) None of these
Solution:
(a) A is singular
\[A+A^T=0\]
\[A^T=-A\]
\[ \left | A^T \right |=\left | -A \right |\]
\[2\left | A \right |=0\]
\[\left | A\right |=0\]
\[A\text{ is singular}\]
0 Comments