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The rank of the following ( n + 1 ) x ( n + 1) matrix, where a is a real number is
All the rows of the given matrix is same. So the matrix has only one independent row.
Rank of the matrix = Number of independent rows of the matrix.
∴ Rank of given matrix = 1
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?
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).
The matrices commute under multiplication
Consider the following set of equations:
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15
Set of equations is
Above set of equations can be written as
Augmented matrix [AB] is given as
Performing gauss-Elimination on the above matrix
An n x n array v is defined as follows:
The sum of the elements of the array v is
The matrix V can be defined as
So above is antisymmetric matrix and the sum of the elements of any antisymmetric matrix is 0.
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?
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.
Let A, B, C, D be n x n matrices, each with non-zero determinant, If ABCD = I, then B-1 is
A, B, C, D is n x n matrix.
Given, ABCD = I
What values of x, y and z satisfy the following system of linear equations?
How many solutions does the following system of linear equations have?
- x + 5y = - 1
x - y = 2
x + 3y = 3
The augmented matrix is
Using gauss-efimination on above matrix we get,
Rank [A | B] - 2 (number of non zero rows in [A | B])
Rank [A ] = 2 (number of non zero row s in [A ])
Rank [A | B] = Rank [A] = 2 = number of variables
∴ Unique solution exists.
Consider the following system of equations in three real variables x1, x2 and x3
This system of equations has
The augmented matrix for the given system is
Using gauss -elimination method on above matrix we get,
Since Rank ([A | B]) = Rank ([A]) = number of variables, the system has unique solution.
What are the eigenvalues of the following 2 x 2 matrix?
The characteristic equation of this matrix is given by
∴ The eigen values of A are 1 and 6.
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?
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.
What are the eigenvalues of the matrix P given below:
Since P is a square matrix
Sum (eigen values) = Trace (P)
= a + a + a = 3a
Product of eigen values
Only choice (a) has sum of eigen values = 3a and product of eigen values = a3 - 2a.
If the rank of a (5 x 6) matrix Q is 4, then which one of the following statements is correct?
If rank of (5 x 6) matrix is 4, then surely it must have exactly 4 linearly independent rows as will as 4 linearly in dependent columns.
The trace and determinant of a 2 x 2 matrix are known to be -2 and -35 respectively. It eigenvalues are
Trace = Sum of principal diagonal element.