Test: Linear Algebra - Electrical Engineering (EE)
15 Questions MCQ Test GATE Electrical Engineering (EE) 2024 Mock Test Series - Test: Linear Algebra
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Let AX - B be a system of linear equations where A is an m x n matrix and b is a m x 1 column vector and X is an x 1 column vector of unknown. Which of the following is false?
Detailed Solution for Test: Linear Algebra - Question 2
Following are the possibilities for a system of linear equations:
(i) If matrix A and augmented matrix [AB] have same rank, then the system has solution otherwise there is no solution.
(ii) If matrix A and augmented matrix [AB] have same rank which is equal to the number of variables, then the system has unique solution and if B is zero vector then the system have only a trivial solution.
(iii) If matrix A and matrix [AB] have same rank which is less than the number of variables, then the system has infinite solution.
Therefore, option (c) is false because if m - n and B is non-zero vector, then it is not necessary that system has a unique solution, because m is the number of equations (quantity) and not the number of linearly independent equations (quality).
Consider the following statements:
S1: The sum of two singular n x n matrices may be non-singular
S2: The sum of two n x n non-singular matrices may be singular
Which of the following statements is correct?
Detailed Solution for Test: Linear Algebra - Question 6
S1 is true
Consider two singular matrices
Sum of A and B is given as
However (A + B) is a non-singuiar matrix
So, S1 is true.
Now, consider two non-singuiar matrices
However (C + D) is a singular matrix. So, S2 is also true.
Therefore, both S1 and S2 are true.
F is an n x n real matrix, b is an n x 1 real vector. Suppose there are two n x 1 vectors, u and v such that u ≠ v and Fu = b, Fv = b.
Which one of the following statements is false?
Detailed Solution for Test: Linear Algebra - Question 12
Given that Fu= b and Fv= b
If F is non singular, then it has a unique inverse.
Now, u = F-1 b and v = F-1 b
Since F-1 is unique u = v but it is given that u ≠ v .
This is a contradiction. So F must be singular.
This means that
(a) Determinate of F is zero is true. Also
(b) There are infinite number of solution to Fx = b is true since |F| = 0
(c) There is an X ≠ 0 such that FX = 0 is also true, since X has infinite number of solutions, including the X = 0 solution.
(d) F must have 2 identical rows is false, since a determinant may become zero, even if two identical columns are present. It is not necessary that 2 identical rows must be present for |F| to become zero.
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