Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) PDF Download

Q26: A system matrix is given as follows
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is _______.      (SET-1 (2014))
(a) 1
(b) 2
(c) 3
(d) 4
Ans:
(c)
Sol: Characteristic equation ∣A−λI∣= 0
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The absolute values of λ are ∣𝜆∣ = 1, 2, 3
 Ratio of maximum and minimum eigen value is = 3 : 1 = 3/1 = 3

Q27: Given a system of equations
x + 2y + 2z = b1
5x + y + 3z = b2
What of the following is true regarding its solutions      (SET-1(2014))
(a) The system has a unique solution for any given b1 and b2
(b) The system will have infinitely many solutions for any given b1 and b2
(c) Whether or not a solution exists depends on the given b1 and b2
(d) The systems would have no solution for any values of b1 and b2
Ans: 
(b)

Q28: A Matrix has eigenvalues -1 and -2. The corresponding eigenvectors are Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) respectively. The matrix is     (2013)
(a) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: d
Sol: AX = λX
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)From equation (i) and (iii), a = 0 and b = 1
From equation (ii) and (iv), c = -2 and d = -3
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q29: The equation Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) has      (2013)
(a) no solution
(b) only one solution Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(c) non-zero unique solution
(d) multiple solutions
Ans:
(d)
Sol: Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)i.e. 𝑥1x1 and x2 are having infinite number of solutions. 
⇒ Multiple solutions are these.

Q30: Given that Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) the value of A3 is      (2012 )
(a) 15A + 12I
(b) 19A + 30I
(c) 17A + 15I
(d) 17A + 21I

Ans: (b) 
Sol: Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Characteristic equation of A is
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(by Cayley Hamilton theorem)
⇒ A= −5A − 6I
Multiplying by A on both sides, we have,
A3= −5A− 6A  
⇒ A3 =−5(−5A−6I)−6A = 19A + 30I

Q31: The matrix Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are     (2011)
(a) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (d)
Sol: Let us try Dolittle's decomposition by putting 𝑙11 = 1 and 𝑙22 = 1
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)So one possible breakdown is
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)But this is not any of the choises given.
So let us do Crout's decomposition, by putting 𝑢11=1u11 = 1 and u22 = 1
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q32:  For the set of equations, 𝑥1+2𝑥2+𝑥3+4𝑥4=2x+ 2x+ x+ 4x= 2 and 3𝑥+ 6𝑥+ 3𝑥+ 12𝑥= 6The following statement is true.      (2010)
(a) Only the trivial solution 𝑥1 = 𝑥2 = 𝑥3 = 𝑥4 = 0 exists
(b) There are no solutions
(c) A unique non-trivial solution exists
(d) Multiple non-trivial solutions exist
Ans:
(d)
Sol: Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The augmental matrix is Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Performing gauss-elimination on this we get
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)rank(A) = rank(A∣B) = 1
So, system is consistent.
Since, system's rank = 1 is less than the number of variables, only infinite (multiple) non-trival solution exists.

Q33: An eigenvector of Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is       (2010)
(a) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (b)
Sol: Given,
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
P is triangular. So eigen values are the diagonal elements themselves. Eigen values are therefore, 𝜆1=1,𝜆2=2,𝜆3=3.λ= 1, λ= 2, λ= 3. 
Now, the eigen value problem is [𝐴𝜆𝐼]𝑥^=0[A − λI] Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Putting λ1 = 1, we get the eigen vector corresponding to this eigen value,
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which gives the equations
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)The solution is 𝑥= 0, 𝑥= 0, 𝑥1= 𝑘
So, one eigen vector is Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Since, none of the eigen vectors given in choises matches with this, ratio we need to proceed further and find the other eigen vectors corresponding to the other eigen values.
Now, corresponding to 𝜆2=2,λ= 2, we get by substituting 𝜆=2,λ 2in the eigen value problem, the following set of equations, 
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which gives the equations,
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Solution is 𝑥3=0,𝑥1=𝑘,𝑥2=𝑘x= 0, x= k, x= k
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Since none of the eigen vectors given in the choises is of this ratio, we need to proceed further and find 3rd eigen vector also.
By putting 𝜆=3,λ 3, in the eigen value problem, we get 
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Only the eigen vector geven in choise Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is in this ratio.

Q34: The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. Its eigenvalues are     (2009)
(a) -30 and -5
(b) -37 and -1
(c) -7 and 5
(d) 17.5 and -2
Ans:
(c)
Sol: Trace = Sum of principal diagonal element.

Q35: Let P be a 2x2 real orthogonal matrix and Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is a real vector  Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) Then, which one of the following statements is correct?    (2008)
(a) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) No relationship can be established between Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (b)
Sol: Let on orthogonal matrixPrevious Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Properly of orthogonal matrix A
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q36: A is  m × n full rank matrix with m > n and I is identity matrix. Let matrix A+ = (ATA)−1ATThen, which one of the following statement is FALSE ?    (2008)
(a) 𝐴𝐴+𝐴=𝐴AA+A =  A  
(b) (AA+)2  = AA+
(c) A+A = I
(d) AA + A = A +
Ans: 
(b)
Sol: Choice (A) AA+A = A is corect
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q37: If the rank of a (5 x 6) matrix Q is 4, then which one of the following statement is correct ?    (2008)
(a) Q will have four linearly independent rows and four linearly independent columns
(b) Q will have four linearly independent rows and five linearly independent columns
(c) QQT will be invertible
(d) QTQ will be invertible
Ans: 
(a)
Sol: If rank of (5 x 6) matrix is 4, then surely it must have exactly 4 linearly independent rows as will as 4 linearly independent columns.

Q38: The characteristic equation of a (3x3) matrix P is defined as
a(λ)∣λI−P∣ = λ32+2λ+1 = 0
If I denotes identity matrix, then the inverse of matrix P will be     (2008)
(a) (𝑃2+𝑃+2𝐼)(P2+P+2I)
(b) (P2+P+I)
(c) −(P2+P+I)
(d) −(P2+P+2I)
Ans:
(d)
Sol: If characteristic equation is
λ+ λ+ 2λ + 1 = 0
Then by cayley- hamilton theorem,
P+ P+ 2P + I = 0
I = −P− P− 2P
Multiplying by P−1 on both sides,
𝑃−1 = −𝑃− 𝑃 − 2𝐼 = −(𝑃+ 𝑃 + 2𝐼)  

Q39: Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) A9 equals      (2007)
(a) 511𝐴+510𝐼511A + 510I
(b) 309𝐴+104𝐼309A + 104I
(c) 154A + 155I
(d) exp(9A)
Ans:
a
Sol: To calculate A9
start from A+ 3A + 2I = 0 which has been derived above
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q40: Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)A satisfies the relation      (2007)
(a) A + 3I + 2A−1 = 0
(b) A+ 2A + 2I = 0
(c) (A + I)(A + 2I)
(d) exp(A) = 0
Ans: 
(a)
Sol: Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)A will satisfy this equation according to cayley hamition theorem
i.e. A+ 3A + 2I = 0
multiplying by A−1 on both sides we get
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q41: The linear operation L(x) is defined by the cross product Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)are three dimensional vectors. The 3 x 3 matrix M of this operations satisfies
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Then the eigenvalues of M are     (2007)
(a) 0, +1, -1
(b) 1, -1, 1
(c) i, -i, 1
(d) i, -i, 0
Ans:
(c)
Sol: The eigen values of M are (i, −i, 1)

Q42: Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant       (2007)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)(a) is zero when x and y are linearly independent
(b) is positive when x and y are linearly independent
(c) is non-zero for all non-zero x and y
(d) is zero only when either x or y is zero
Ans:
(b)

Q43: x = [x1 x2 ... xn]T is an n-tuple nonzero vector. The nxn matrix V = xxT       (2007)
(a) has rank zero
(b) has rank 1
(c) is orthogonal
(d) has rank n
Ans:
(b)
Sol: The nxn matrix, V = XXT has rank 1.

Q44: If Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)then top row of R−1 is      (2005)
(a) [5   6   4]
(b) [5   -3 1]
(c) [2  0  -1]
(d) [2  1   1/2]
Ans: 
(b)
Sol:
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Since we need only the top row of 𝑅1R−1, we need to find only first column of (R) which after transpose will become first row of Adj(A). 
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Q45: For the matrix Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)one of the eigen values is equal to -2. Which of the following is an eigen vector ?      (2005)
(a) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)

(b)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(c)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
(d) Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)
Ans: (d)
Sol: Since matrix is triangular, the eigen values are the diagonal elements themselves namely λ = 3,−2 and 1.
Corresponding to eigen value λ = −2 let us find the eigen vector
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Which gives the equations,
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Since equation (ii) and (iii) are same we have
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Eigen vectors are of the form
Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE)Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is an eigen vector of matrix p.

Q46: In the matrix equation Px = q, which of the following is a necessary condition for the existence of at least on solution for the unknown vector x:     (2005)
(a) Augmented matrix [Pq] must have the same rank as matrix P
(b) Vector q must have only non-zero elements
(c) Matrix P must be singular
(d) Matrix P must be square
Ans:
(a)
Sol: rank [Pq] = rank [P] is necessary for existence of at least one solution to Px = q.

Q47: The eigen values of the system represented by Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) X are     (2002)
(a) 0, 0, 0, 0
(b) 1, 1, 1, 1
(c) 0, 0, 0, -1
(d) 1, 0, 0, 0
Ans:
(d)

Q48: The determinant of the matrix Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is:     (2002)
(a) 100
(b) 200
(c) 1
(d) 300
Ans: 
(c)
Sol: The given matrix is alower traingular matrix
Therefore, |A|= Product of its leading diagonal elemetns = (1)(1)(1)(1) = 1

The document Previous Year Questions- Linear Algebra - 2 | Engineering Mathematics for Electrical Engineering - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Engineering Mathematics for Electrical Engineering.
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1. What are the basic operations in linear algebra?
Ans. The basic operations in linear algebra include addition, subtraction, scalar multiplication, matrix multiplication, and finding the inverse of a matrix.
2. How is linear algebra used in electrical engineering?
Ans. Linear algebra is used in electrical engineering to analyze circuits, control systems, signal processing, and communication systems. It helps in solving equations with multiple variables and understanding the behavior of complex systems.
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Ans. Eigenvalues and eigenvectors are important concepts in linear algebra as they help in understanding the behavior of linear transformations. They are used in various applications such as solving differential equations, image processing, and machine learning algorithms.
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