Competitive Linear Algebra Overview MCQs ( Signals and Systems ) MCQs – Signals and Systems MCQs

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Signals and Systems MCQs – Linear Algebra Overview MCQs ( Signals and Systems ) MCQs

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Basics of Linear Algebra

1. Find the values of x, y, z and w from the below condition.
5[xyzw]=[23102x+y]+[z75w].
a) x=1, y=3, z=4, w=0
b) x=2, y=3, z=8, w=1
c) x=1, y=2, z=3, w=1
d) x=1, y=2, z=4, w=1
Answer: d
Explanation: 5z=10+5 => 5z=15 => z=3
5x=2+z => 5x=5 => x=1
5y=3+7 => 5y=10 => y=2
5w=2+2+w => 4w=4 => w=1.


2. The matrix A is represented as ⎡⎣⎢1−2−349−8⎤⎦⎥. The transpose of the matrix of this matrix is represented as?
a) [1−249]
b) ⎡⎣⎢1−2−3498⎤⎦⎥
c) [14−29−38]
d) [−1−42−938]
Answer: c
Explanation: Given matrix is a 3×2 matrix and the transpose of the matrix is 3×2 matrix.
The values of matrix are not changed but, the elements are interchanged, as row elements of a given matrix to the column elements of the transpose matrix and vice versa but the polarities of the elements remains same.


3. Find the inverse of the matrix A=⎡⎣⎢847564232⎤⎦⎥.
a) 113∗⎡⎣⎢906580656158805469⎤⎦⎥
b) 114∗⎡⎣⎢936880686158805869⎤⎦⎥
c) 113∗⎡⎣⎢946780676058805669⎤⎦⎥
d) 113∗⎡⎣⎢936880686158805869⎤⎦⎥
Answer: d
Explanation: The inverse of matrix A = adjA|A|,
adjA=AA^-1,
adjA = 113∗⎡⎣⎢936880686158805869⎤⎦⎥, |A|=13.


4. Given the equations are 4x+2y+z=8, x+ y+ z=3, 3x+y+3z=9. Find the values of x, y and z.
a) 5/3, 0, 2/3
b) 1, 2, 3
c) 4/3, 1/3, 5/3
d) 2, 3, 4
Answer: a
Explanation: The matrix from the equations is represented as M=⎡⎣⎢413211113⎤⎦⎥
The another matrix is X = ⎡⎣⎢839⎤⎦⎥
Then |M| = 6
For x=⎡⎣⎢839211113⎤⎦⎥ = 5/3
Similarly, y=0, z=-2/3.


5. Find the adjacent A as A=⎡⎣⎢156748−3−2−6⎤⎦⎥.
a) ⎡⎣⎢112123134⎤⎦⎥
b) ⎡⎣⎢3139803945748074136⎤⎦⎥
c) ⎡⎣⎢100010001⎤⎦⎥
d) ⎡⎣⎢355698346774324852⎤⎦⎥
Answer: b
Explanation: The adjacency of A is given by AA^T
A^T = ⎡⎣⎢17−354−268−6⎤⎦⎥,
AA^T = ⎡⎣⎢156748−3−2−6⎤⎦⎥×⎡⎣⎢17−354−268−6⎤⎦⎥
adjA=⎡⎣⎢3139803945748074136⎤⎦⎥.


6. Find the rank of the matrix A=⎡⎣⎢141362572⎤⎦⎥.
a) 3
b) 2
c) 1
d) 0
Answer: a
Explanation: To find out the rank of the matrix first find the |A|
If the value of the|A| = 0 then the matrix is said to be reduced
But, as the determinant of A has some finite value then, the rank of the matrix is 3.


7. The rank of the matrix (m × n) where m<n cannot be more than?
a) m
b) n
c) m*n
d) m-n
Answer: a
Explanation: let us consider a 2×3 matrix [141516]
Where R₁≠R₂ rank is 2
Another 2×3 matrix [111111]
Here, R₁=R₂ rank is 1
And the rank of these two matrices is 1, 2
So rank is cannot be more than m.


8. Given A=[20−0.13]A−1=[1/20ab] then find a + b.
a) 620
b) 720
c) 820
d) 520
Answer: b
Explanation: AA^-1 = I = [102−0.1b3b]=[1001]
Therefore, a = 160 and b = 13 and a + b = 720.


9. If a square matrix B is skew symmetric then.
a) B^T = -B
b) B^T = B
c) B^-1 = B
d) B^-1 = B^T
Answer: a
Explanation: The transpose of a skew symmetric matrix should be equal to the negative of the matrix
Example: let us consider a matrix B = ⎡⎣⎢a−e−deb−fdfc⎤⎦⎥, B^T = ⎡⎣⎢aed−ebf−d−fc⎤⎦⎥.

Three Domain Representation For LTI Systems MCQs

10. For the following set of simultaneous equations 1.5x-0.5y=2, 4x+2y+3z=9, 7x+y+5=10.
a) The solution is unique
b) Infinitely many solutions exist
c) The equations are incompatible
d) Finite number of multiple solutions exist
Answer: a
Explanation: The equations can be written as ⎡⎣⎢1.547−0.521035⎤⎦⎥
It can also be written as A = ⎡⎣⎢347−221035⎤⎦⎥, |A|=19
Hence, it has a unique solution.

Eigenvalues

1. Find the Eigen values of matrix A=⎡⎣⎢210121012⎤⎦⎥.
a) 2 + 2–√, 2-2–√, 2
b) 2, 1, 2
c) 2, 2, 0
d) 2, 2, 2
Answer: a
Explanation: To find the Eigen values it satisfy the condition, |A-λI|=0
|A-λI| = ⎡⎣⎢210121012⎤⎦⎥–λ⎡⎣⎢100010001⎤⎦⎥
|A-λI| = ∣∣∣∣2−λ1012−λ1012−λ∣∣∣∣
= 2 – (λ²-4λ+3) – (2-λ)
By solving the above equation, we get,
λ = 2 + 2–√, 2-2–√, 2.


2. Find the product of Eigen values of a matrix A=⎡⎣⎢103261402⎤⎦⎥.
a) 60
b) 45
c) -60
d) 40
Answer: c
Explanation: According to the property of Eigen values, the product of the Eigen values of a given matrix is equal to the determinant of the matrix |A| = 1(12-0) – 2(0) + 4(8)
= -60.


3. Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix A^T could be.
a) a, b, c
b) -a, -b, -c
c) a-b, b-a, c-a
d) a^-1, b^-1, c^-1
Answer: a
Explanation: According to the property of the Eigen values, any square matrix A and its transpose A^T have the same Eigen values.


4. What is Eigen value?
a) A vector obtained from the coordinates
b) A matrix determined from the algebraic equations
c) A scalar associated with a given linear transformation
d) It is the inverse of the transform
Answer: c
Explanation: Eigen values is a scalar associated with a given linear transformation of a vector space and having the property that there is some nonzero vector which is when multiplied by the scalar is equal to the vector obtained by letting the transformation operate on the vector.


5. Find the sum of the Eigen values of the matrix A=⎡⎣⎢357649721⎤⎦⎥.
a) 7
b) 8
c) 9
d) 10
Answer: b
Explanation: According to the property of the Eigen values, the sum of the Eigen values of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigen values = 3 + 4 + 1 = 8.


6. Let the matrix A be the idempotent matrix then the Eigen values of the idempotent matrix are ________
a) 0, 1
b) 0
c) 1
d) -1
Answer: a
Explanation: According to the property of the Eigen values, the Eigen values of the idempotent matrix are either zero or unity.
So, the answer is 0 or 1.


7. Let us consider a 3×3 matrix A with Eigen values of λ₁, λ₂, λ₃ and the Eigen values of A^-1 are?
a) λ₁, λ₂, λ₃
b) 1λ1,1λ2,1λ3
c) -λ₁, -λ₂, -λ₃
d) λ₁, 0, 0
Answer: b
Explanation: According to the property of the Eigen values, if is the Eigen value of A, then 1λ is the Eigen value of A^-1.
So the Eigen values of A^-1 are 1λ1,1λ2,1λ3.


8. The Eigen values of a 3×3 matrix are λ₁, λ₂, λ₃ then the Eigen values of a matrix A³ are __________
a) λ₁, λ₂, λ₃
b) 1λ1,1λ2,1λ3
c) λ31,λ32,λ33
d) 1, 1, 1
Answer: c
Explanation: If λ₁, λ₂, λ₃……… λ_n are the Eigen values of matrix A then the Eigen values of matrix A^m are said to be λm1,λm2,λm3,………λmn.
So, the answer is λ31,λ32,λ33.


9. Find the Eigen values of matrix A=[4114].
a) 3, -3
b) -3, -5
c) 3, 5
d) 5, 0
Answer: c
Explanation: According to the property of the Eigen value, the eigen values are determined as follows:
4 + 4 = 8
3 + 5 = 8
The sum of the Eigen values is equal to the sum of the principal diagonal elements of the matrix.


10. Where do we use Eigen values?
a) Fashion or cosmetics
b) Communication systems
c) Operations
d) Natural herbals
Answer: b
Explanation: Eigen values are used in communication systems, designing bridges, designing car stereo system, electrical engineering, mechanical companies.

Competitive Linear Algebra Overview MCQs ( Signals and Systems ) MCQs – Signals and Systems MCQs