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 nonzero 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
and
Consider the following set of equations:
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15
This set
Set of equations is
Above set of equations can be written as
Augmented matrix [AB] is given as
Performing gaussElimination 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.
OR
Alternate Method:
Consider the following statements:
S_{1}: The sum of two singular n x n matrices may be nonsingular
S_{2}: The sum of two n x n nonsingular matrices may be singular
Which of the following statements is correct?
S_{1} is true
Consider two singular matrices
Sum of A and B is given as
However (A + B) is a nonsinguiar matrix
So, S_{1} is true.
Now, consider two nonsinguiar matrices
However (C + D) is a singular matrix. So, S_{2} is also true.
Therefore, both S_{1} and S_{2} are true.
Let A, B, C, D be n x n matrices, each with nonzero 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 gaussefimination 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 x_{1}, x_{2} and x_{3}
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 = a^{3}  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.
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