Classical

Linear Algebra - Linear Algebra

6:  

 If A and B are square matrices of size n x n, then which of the following statement is not true? 

 
A.

det. (AB) = det (A) det (B)

B.

det (kA) = kn det (A)

C.

det (A + B) = det (A) + det (B)

D.

det (AT) =1/det (A-1)

 
 

Option: C

Explanation :


7:  

 In the matrix equation Px = q. which of the following is a necessary condition for the existence of at least one solution for the unknown vector x?

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. None of these
 
 

Option: A

Explanation :
According to Rouche's theorem, the system is consistent if and only if the coefficient matrix and the augmented matrix K are of the same rank, otherwise the system is inconsistent.


8:  

Matrix, A =

 
cosΘ sinΘ 0
sinΘ cosΘ 0
0 0 1
 
A.

orthogonal

B.

non-singular

C.

have A-1 exists

D.

both (b) & (c)

 
 

Option: D

Explanation :

Determinant A = 1 (cos2Θ + sin2Θ) Hence A is non-singular and A-1 exists


9:  

The matrix B = AT, where A is any matrix is

A. skew symmetric
B. symmetric about the secondary diagonal
C. always symmetric
D. another general matrix
 
 

Option: C

Explanation :


10:  

Eigen values of a square symmetric matrix are always

A. positive
B. real and imaginary
C. negative
D. real
 
 

Option: D

Explanation :




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