Linear Algebra

Question 1
Consider the following system of linear equation.

x_1+2x_2=b_1;
2x_1+4x_2=b_2;
3x_1+7x_2=b_3;
3x_1+9x_2=b_4;

Which one of the following conditions ensures that a solution exists for the above system?
A
b_2=2b_1 and 6b_1-3b_3+b_4=0
B
b_3=2b_1 and 6b_1-3b_3+b_4=0
C
b_2=2b_1 and 3b_1-6b_3+b_4=0
D
b_3=2b_1 and 3b_1-6b_3+b_4=0
GATE EC 2020   Engineering Mathematics
Question 1 Explanation: 
Given:
x_{1}+2x_{2}=b_{1} \; \; ...(i)
2x_{1}+4X_{2}=b_{2}\; \; ...(ii)
3x_{1}+7x_{2}=b_{3}\; \; ...(iii)
3x_{1}+9x_{2}=b_{4}\; \; ...(iv)
From equations (ii) and (i)
We can write,
b_{2}=2[x_{1}+2x_{2}]=2b_{1}
From option (C):
3b_{1}-6b_{3}+b_{4}=3[x_{1}+2x_{2}]-6[3x_{1}+7x_{2}]+3x_{1}+9x_{2}\neq 0
From option (A):
b_{2}=2b_{1}
and b_{1}-3b_{3}+b_{4}=6[x_{1}+2x_{2}]-3[3x_{1}+7x_{2}]+[3x_{1}+9x_{2}]=0
6b_{1}-3b_{3}+b_{4}=0
Hence, answer is option (A).
Question 2
The number of distinct eigenvalues of the matrix
\begin{bmatrix} 2 & 2 & 3 &3 \\ 0& 1 & 1 & 1\\ 0 & 0 & 3 &3 \\ 0&0 & 0 &2 \end{bmatrix}
is equal to ____
A
1
B
2
C
3
D
4
GATE EC 2019   Engineering Mathematics
Question 2 Explanation: 
A=\left[\begin{array}{llll} 2 & 2 & 3 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 2 \end{array}\right]
Eigen values are 2,1,3,2
Distinct eigen values are 2,1,3[/latex]
\therefore Number of distinct eigen values =3
Question 3
Consider matrix A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix} and vector x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}. The number of distinct real values of k for which the equation Ax=0 has infinitely many solutions is _______.
A
1
B
2
C
3
D
4
GATE EC 2018   Engineering Mathematics
Question 3 Explanation: 
A X=0 has infinitely many solutions
\begin{aligned} \text{So, }\quad|A|=0 ;&\left|\begin{array}{cc}k & 2 k \\ k^{2}-k k^{2}\end{array}\right|=0 \\ k^{3}-2 k^{3}+2 k^{2} &=0 ; \quad k^{2}(2-k)=0 \\ k &=0,2 \end{aligned}
\Rightarrow "two" distinct values of k
Question 4
Let M be a real 4\times4 matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
A
S1 implies S2
B
S1 implies S3
C
S2 implies S1
D
S3 implies S2
GATE EC 2018   Engineering Mathematics
Question 4 Explanation: 
Eigen vectors corresponding to distinct eigen values are linearly independent.
Hence, "S2 implies S1 ".
Question 5
The rank of the matrix \begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ -1 & 0& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix} is ___________.
A
4
B
5
C
6
D
3
GATE EC 2017-SET-2   Engineering Mathematics
Question 5 Explanation: 
A=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ -1 & 0& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_1
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & -1& 0& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_3
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & 0& -1& 0 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\rightarrow R_4+R_2
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 0& 1 &-1 &0 \\ 0 & 1 &-1 & 0&0 \\ 0 & 0& 0& -1 &1 \\ 0 & 0 & 0 &1 &-1 \end{bmatrix}
R_4\leftrightarrow R_3 \text{ and }R_5\rightarrow R_5+R_4
=\begin{bmatrix} 1& -1 & 0& 0&0 \\ 0& 1& -1 & &0 \\ 0 & 0 &1 & -1&0 \\ 0 & 0& 0& -1 &1 \\ 0 & 0 & 0 &0 &0 \end{bmatrix}
From here,
\therefore \;\; \rho (A)=4
Question 6
The rank of the matrix M=\begin{bmatrix} 5 & 10 &10 \\ 1& 0 &2 \\ 3 & 6&6 \end{bmatrix} is
A
0
B
1
C
2
D
3
GATE EC 2017-SET-1   Engineering Mathematics
Question 6 Explanation: 
\begin{aligned} &M=\left[\begin{array}{lll} 5 & 10 & 10 \\ 1 & 0 & 2 \\ 3 & 6 & 6 \end{array}\right]\\ &R_{1} \leftrightarrow R_{2}:\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 5 & 10 & 10 \\ 3 & 6 & 6 \end{array}\right]\\ &R_{2} \leftarrow R_{2}-5 R_{1} \text { and } R_{3} \leftarrow R_{3}-3 R_{1}\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 10 & 0 \\ 0 & 6 & 0 \end{array}\right]\\ &R_{3} \leftarrow R_{3}-\frac{6}{10} R_{2}\\ &\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 10 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{aligned}
Which is in Echelon form
Rank of matrix M is,
\rho (M) = 2
Question 7
Consider the 5 x 5 matrix
A=\begin{bmatrix} 1 &2 &3 & 4 & 5\\ 5 & 1& 2& 3& 4\\ 4& 5&1 & 2&3 \\ 3 & 4 & 5 & 1 & 2\\ 2&3 & 4&5 & 1 \end{bmatrix}
It is given that A has only one real eigen value. Then the real eigen value of A is
A
-2.5
B
0
C
15
D
25
GATE EC 2017-SET-1   Engineering Mathematics
Question 7 Explanation: 
\begin{aligned} |A-\lambda I|&=0 \\ \begin{array}{|ccccc|} 1-\lambda & 2 & 3 & 4 & 5 \\ 5 & 1-\lambda & 2 & 3 & 4 \\ 4 & 5 & 1-\lambda & 2 & 3 \\ 3 & 4 & 5 & 1-\lambda & 2 \\ 2 & 3 & 4 & 5 & 1-\lambda \end{array}&=0\\ \end{aligned}
sum of all elements in every one row must be zero.
\begin{aligned} \text{i.e. }\quad 15-\lambda&=0\\ \lambda&=15 \end{aligned}
Question 8
Consider a 2 x 2 square matrix

A=\begin{bmatrix} \sigma & x\\ \omega & \sigma \end{bmatrix}

where x is unknown. If the eigenvalues of the matrix A are (\sigma +j\omega ) \; and \; (\sigma -j\omega ), then x is equal to
A
+j\omega
B
-j\omega
C
+\omega
D
-\omega
GATE EC 2016-SET-3   Engineering Mathematics
Question 8 Explanation: 
\begin{aligned} A &=\left[\begin{array}{ll} \sigma & x \\ \omega & \sigma \end{array}\right] \\ \text { Trace } &=\text { sum of eigen values } \\ 2 \sigma &=\sigma+j \omega+\sigma-j \omega \\ |A| &=\text { product of eigens } \\ \sigma^{2}-x \omega &=(\sigma+j \omega)(\sigma-j \omega)=\sigma^{2}+\omega^{2} \end{aligned}
which is possible only when, x=-\omega
Question 9
The matrix \begin{bmatrix} a & 0&3 & 7\\ 2& 5& 1 &3 \\ 0 & 0 & 2& 4\\ 0 &0 & 0&b \end{bmatrix} has det(A) = 100 and trace(A) = 14.

The value of |a-b| is ________
A
1
B
2
C
3
D
4
GATE EC 2016-SET-2   Engineering Mathematics
Question 9 Explanation: 
\begin{aligned} \text { Trace of } A&=14 \\ 2+b&=14 \\ a+b &=7 &\ldots(i)\\ \text { det }(A)&=100 \\ \begin{array}{r|rrr|} & a & 37 & \\5& 0 & 2 & 4 \\& 0 & 0 & b\end{array}=100\\ 5 \times 2 \times a \times b &=100 \\ 10 a b &=100 \\ a b &=10 &\ldots(ii) \end{aligned}
From equation (i) and (ii)
\begin{aligned} \text { either }\quad a&=5,b=2\\ \text{or}\quad a&=2b=5\\ |a-b|&=|5-2|=3 \end{aligned}
Question 10
The value of x for which the matrix A=\begin{bmatrix} 3 & 2&4 \\ 9& 7&13 \\ -6&-4 & -9+x \end{bmatrix}
has zero as an eigenvalue is ________
A
0
B
1
C
2
D
3
GATE EC 2016-SET-2   Engineering Mathematics
Question 10 Explanation: 
A has an eigen value is zero
\begin{aligned} \therefore \quad|A|&=0 \\ \left| \begin{array}{lll} 3 & 2 & 4\\9 & 7 & 13 \\ -6 & -4 & -9+x\end{array}\right|&=0 \\ 3(-63+7 x+52)-2(-81+9 x+78)&+4(-36+42)=0 \\ 3(7 x-11)-2(9 x-3)+4(6)&=0 \\ 21 x-33-18 x+6+24&=0 \\ 3 x-3&=0 \\ x&=1 \end{aligned}
There are 10 questions to complete.
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