Linear algebra covers almost 5-6 marks in GATE exam which makes it very important from a scoring perspective. Try out these questions on solutions on linear equations and eigen values and eigen vectors and check how much you understand the concepts involved.

Q1) The system of linear equations

has

1. a unique solution
2. no solution
3. an infinite number of solutions
4. exactly two distinct solutions

Q2) Solve the following set of equations

Q3) For the following equations:

1. the solution is unique
2. infinitely many solutions exist
3. the equations are incompatible
4. finite number of multiple solutions exist

Q4) The following set of equations

has

1. no solution
2. a unique solution
3. multiple solution
4. an inconsistent solution

Q5) For what value of 'a', if any, will the following system of equations in x, y and z will have a solution?

1. any real number
2. 0
3. 1
4. there is no such value

Q6) Solution for the system defined by the set of equations:

is

1. x=0, y=1, z=4/3
2. x=0, y=1/2, z=2
3. x=1, y=1/2, z=2
4. non-existent

Q7) For what values of  and  the following simultaneous equations have an infinite number of solutions?

1. 2,7
2. 3,8
3. 8,3
4. 7,2

Q8) Let A be a 3x3 matrix with rank 2. Then AX=0 has

1. only the trivial solution X=0
2. one independent solution
3. two independent solutions
4. three independent solutions

Q9) For the matrix , the eigen value corresponding to the eigenvector  is

1. 2
2. 4
3. 6
4. 8

Q10) The three characteristic roots of the following matrix A =  are

1. 2,3
2. 1,2,2
3. 1,0,0
4. 0,2,3

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Sol1) Write the given system as AX=B, where A =

The augmented matrix is

Rank of augmented matrix =

So  which means that the system is inconsistent, so no solution.

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Nice work guys!

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