Linear Algebra MCQ

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Let A be an m × n matrix and Ban n × m matrix. It is given that determinant ($$I$$m + AB) = determinant ($$I$$n + BA), where $$I$$k is the k × k identity matrix. Using the above property, the determinant of the matrix given below is \[\left[ {\begin{array}{*{20}{c}} 2&1&1&1 \\ 1&2&1&1 \\ 1&1&2&1 \\ 1&1&1&2 \end{array}} \right

ক
\

খ
A. 2

গ
 5
For the set of equations x1 + 2x2 + x3 + 4x4 = 2 3x1 + 6x2 + 3x3 + 12x4 = 6 the following statement is true:

ক
Only the trivial solution x1 = x2 = x3 = x4 = 0 exists

খ
There are no solution

গ
A unique non-trivial solution exists

ঘ
Multiple non-trivial solutions exist
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 4&{ - 2} \\ { - 2}&1 \end{array}} \right

ক
\

খ
A. are 1 and 4

গ
 are -1 and 2
Eigen values of a matrix \[{\text{S}} = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right

ক
\

খ
are 5 and 1. What are the eigen values of the matrix S2 = SS?

গ
A. 1 and 25

ঘ
 6 and 4
The smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} 3&{ - 2}&2 \\ 4&{ - 4}&6 \\ 2&{ - 3}&5 \end{array}} \right

ক
\

খ
A. 1.5 and 2.5

গ
 0.5 and 2.5
Let M be a real 4 × 4 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?

ক
S1 implies S2

খ
S1 implies S3

গ
S2 implies S1

ঘ
S3 implies S2
Consider the matrix \[\left[ {\begin{array}{*{20}{c}} 5&{ - 1} \\ 4&1 \end{array}} \right

ক
\

খ
. Which one of the following statements is TRUE for the eigen values and eigen vectors of this matrix?

গ
A. Eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists

ঘ
 Eigen value 3 has a multiplicity of 2, and two independent eigen vector exists
The minimum and the maximum eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&1&3 \\ 1&5&1 \\ 3&1&1 \end{array}} \right

ক
\

খ
are -2 and 6, respectively. What is the other eigen value?

গ
A. 5

ঘ
 3
A is m × n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A' = (ATA)-1AT, Then, which one of the following statement is TRUE?

ক
AA' A = A

খ
(AA')2 = A

গ
AA'A = $$I$$

ঘ
AA'A = A'
X and Y are non-zero square matrices of size n × n. If XY = 0n × n then

ক
|X| = 0 and |Y| ≠ 0

খ
|X| ≠ 0 and |Y| = 0

গ
|X| = 0 and |Y| = 0

ঘ
|X| ≠ 0 and |Y| ≠ 0
The dimension of the null space of the matrix \[\left[ {\begin{array}{*{20}{c}} 0&1&1 \\ 1&{ - 1}&0 \\ { - 1}&0&1 \end{array}} \right

ক
\

খ
is

গ
A. 0

ঘ
 1
Consider a non-homogeneous system of linear equations representing mathematically an overdetermined system. Such a system will be

ক
consistent having a unique solution

খ
consistent having many solutions

গ
inconsistent having a unique solution

ঘ
inconsistent having no solution
The product of eigen values of the matrix P is \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 2&0&1 \\ 4&{ - 3}&3 \\ 0&2&{ - 1} \end{array}} \right

ক
\

খ
A. -6

গ
 2
For the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 5&3 \\ 1&3 \end{array}} \right

ক
,\

খ
ONE of the normalized eigen vectors is given as

গ
A. \

ঘ
 \
The condition for which the eigen values of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&1 \\ 1&{\text{k}} \end{array}} \right

ক
\

খ
are positive, is

গ
A. k > \

ঘ
 k > -2
Given a system of equations: x + 2y + 2z = b1 5x + y + 3z = b2 Which of the following is true regarding its solution?

ক
The system has a unique solution for any given b1 and b2

খ
The system will have infinitely many solutions for any given b1 and b2

গ
Whether or not a solution exists depends on the given b1 and b2

ঘ
The system would have no solution for any values of b1 and b2
For any real, square and non-singular matrix B, the detB-1 is

ক
zero

খ
(detB)-1

গ
- (detB)

ঘ
detB
Which one of the following statements is NOT true for a square matrix A?

ক
If A is upper triangular, the eigen values of A are the diagonal elements of it

খ
If A is real symmetric, the eigen values of A are always real and positive

গ
If A is real, the eigen values of A and AT are always the same

ঘ
If all the principal minors of A are positive, all the eigen values of A are also positive
The trace and determinant of a 2 × 2 matrix are known to be -2 and -35 respectively. It eigen values are

ক
-30 and -5

খ
-37 and -1

গ
-7 and 5

ঘ
17.5 and -2
Let A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE?

ক
Rank of ATA is less than 2

খ
Rank of ATA is equal to 2

গ
Rank of ATA is greater than 2

ঘ
Rank of ATA can be any number between 1 and 3
The product of matrices (PQ)-1P is

ক
P-1

খ
Q-1

গ
P-1Q-1 P

ঘ
PQ P-1
M is a 2 × 2 matrix with eigen values 4 and 9. The eigen values of M2 are

ক
-2 and -3

খ
2 and 3

গ
4 and 9

ঘ
16 and 81
Consider the following system of linear equations \[\left[ {\begin{array}{*{20}{c}} 2&1&{ - 4} \\ 4&3&{ - 12} \\ 1&2&{ - 8} \end{array}} \right

ক
\left = \left\

খ
 Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of \, does this system of equations have infinitely many solutions?

গ
A. 0

ঘ
 1
Consider the systems, each consisting of m linear equations in n variables. I. If m II. If m > n, then none of these systems has a solution. III. If m = n, then there exists a system which has a solution. Which one of the following is CORRECT?

ক
I, II and III are true

খ
Only II and III are true

গ
Only III is true

ঘ
None of them is true
Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that PX = 0 and PY = 0. The dimension to the range space of P is

ক
0

খ
1

গ
2

ঘ
3
x + 2y + z = 4 2x + y + 2z = 5 x - y + z = 1 The system of algebraic given below has

ক
A unique solution of x = 1, y = 1 and z = 1

খ
only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0)

গ
infinite number of solutions

ঘ
no feasible solution
Which one of the following statements is true for all real symmetric matrices?

ক
All the eigen values are real

খ
All the eigen values are positive

গ
All the eigen values are distinct

ঘ
Sum of all the eigen values is zero
The eigen value of the following matrix \[\left[ {\begin{array}{*{20}{c}} {10}&{ - 4} \\ {18}&{ - 12} \end{array}} \right

ক
\

খ
A. 4, 9

গ
 6, -8
For what value of a, if any, will the following system of equations in x, y and z have a solution? 2x + 3y = 4; x + y + z = 4; x + 2y - z = a

ক
Any real number

খ
0

গ
1

ঘ
There is no such value
The system of linear equations 4x + 2y = 7 2x + y = 6 has

ক
a unique solution

খ
no solution

গ
an infinite number of solutions

ঘ
exactly two distinct solutions

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