Linear Algebra - Linear Algebra

51:

The solution of the given matrix equation is

A.	x₁=0, x₂=0, x₃=0
B.	x₁=1, x₂=1, x₃=0
C.	x₁=0, x₂=1, x₃=1
D.	x₁=2, x₂=-5, x₃=-1
	Answer Report Discuss
	Option: A Explanation : Let matrices $A= \begin{bmatrix} 2&2&2\\1&5&-3\\-1&-3&1 \end{bmatrix}\\X = \begin{bmatrix} x_{1}\\x_{2}\\x_{3} \end{bmatrix} and B_{1}=\begin{bmatrix} 0\\0\\0 \end{bmatrix}$ $\therefore$ $Multiplying both sides by A -1$ $X = A -1 B$ $But as B = 0 therfore X = 0$ Hence $\begin{bmatrix} x_{1}\\x_{2}\\x_{3} \end{bmatrix} =\begin{bmatrix} 0\\0\\0 \end{bmatrix}$ $\therefore$ x₁ = 0, x₂ = 0, x₃ = 0 Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

52:

If product of matrix

is a null matrix, then θ and Φ differ by an

A.	odd multiple of π
B.	even multiple of π
C.	odd multiple of π/2
D.	even multiple of π/2
	Answer Report Discuss
	Option: C Explanation : $AB= \begin{bmatrix} cos\theta .cos\phi cos(\theta -\phi)&cos\theta .sin\phi .cos(\theta -\phi)\\cos\phi .sin\theta .cos(\theta -\phi)&sin\theta .sin\phi cos (\theta -\phi ) \end{bmatrix}$ A null matrix, when cos $(\theta -\phi )$ i.e. if $(\theta -\phi )$ is an odd miltiple of $\frac{\pi }{2}$ Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

53:

Sum of the eigen values of the matrix

for real and negative values of x is

A.	greater than zero
B.	less than zero
C.	zero
D.	dependent on value of x
	Answer Report Discuss
	Option: A Explanation : Eigen values are given by the solution of equation $\begin{vmatrix} 3-\lambda &4\\x&1-\lambda \end{vmatrix}=0$ Since x is real and negative, put x = -k, where k is positive constant $\therefore (3 - \lambda) (1 - \lambda) + 4k = 0$ $\Rightarrow \lambda2 - 4\lambda + 3 + 4k = 0$ If λ₁and λ₂ be the solutions of the above equations then λ₁and λ₂are eigen values. Now sum of eigen values = sum of roots of the above equation $i.e. \lambda _{1} + \lambda _{2} = \frac{-(4)}{1}\\$ = 4 (> 0 ) Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

54:

The system of equations

4x + 6y = 8

7x + 8y = 9

3x + 2y = 1

has

A.	no solutions
B.	only one solution
C.	two solutions
D.	infinite number of solutions
	Answer Report Discuss
	Option: B Explanation : For given system of equations $\Delta =\begin{vmatrix} 4&6&8\\7&8&9\\3&2&1 \end{vmatrix}$ = 4 (8 - 18) - 6 (7 - 27) + 8 (14 - 24) = -40 + 120 - 80 = 0 Since $\Delta$ = 0, hence given system of equations has unique solution, i.e. only one solution. Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

55:

The system of equations

4x + 6y = 8

7x + 8y = 9

3x + 2y = 1

has

A.	no solutions
B.	only one solution
C.	two solutions
D.	infinite number of solutions
	Answer Report Discuss
	Option: B Explanation : For given system of equations $\Delta =\begin{vmatrix} 4&6&8\\7&8&9\\3&2&1 \end{vmatrix}$ = 4 (8 - 18) - 6 (7 - 27) + 8 (14 - 24) = -40 + 120 - 80 = 0 Since $\Delta$ = 0, hence given system of equations has unique solution, i.e. only one solution. Click on Discuss to view users comments. Write your comments here: Updating your Comments.. Updating your Comments..

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