Engineering mathematics 1

The system of equations A x =B where A is a m x n singular matrix is consistent if the rank of coefficient matrix A is ----

• equal to the rank of augmented matrix [A:B]

• greater than the rank of [A:B]

• less than the rank of [A:B]

• equal to m x n

The system AX =B where A is a square matrix has unique solution if   $\left|A\right|$    is ---

• zero 

• Not zero

•    $-\left|A\right|$  

• None of these

Let A be 3x3 matrix of rank 2 then the solution of AX = 0 has……

• a trivial solution

• one independent parameter

• two independent parameters

• three independent parameters

The system of equations 2x + 3y = 5 & 4x + 6y = 10 has ----

• no solution

• unique solution

• infinite no. of solutions

• exactly two solutions

If  $X_{1\ \ _{ }}X_{2\ }\ X_3$  are linearly dependent vectors then ----- 

•  $X_1=KX_2\ +lX_3$ 

•  $X_1=X_2\ =X_3$  

•  $X_1=X_2\ =X_3=0$  

• None of these

 $If2,3,4\ are\ the\ eigen\ valuesofmatrixM\ then\left|M\right|=$  

 

• 9

• 24

•  $\frac{1}{9}$  

•  $\frac{1}{24}$  

 $Ifthe\ eigen\ values\ ofa\ matrix\ \ A\ are\ λ_1,\ λ_2\ ,\ λ_3\ then\ eigen\ values\ of\ \ a^{-1}..$  

•  $\frac{1}{\lambda_1}\ \ \frac{1}{\lambda_2}\ \frac{1}{\lambda_3}$  

•  $\lambda_{1\ }\ \lambda_2\ \lambda_3$  

•  $\lambda^1\ \ \lambda^2\ \lambda^3$  

• None of these

If rank of matrix A is r, and number of variables are n then the number of independent solution of the system

AX = 0 is ....

• r

• n

• n-r

• n+r

The characteristic equation of the matrix A is λ² – λ -1 = 0 then

• A^-1 does not exist

• A^-1 exist but cannot be determined

• A^-1 = A - I

• A = A^-1 – I

If the eigen value of a 3x3 matrix A are 2,5,6 the eigen values of     $A^2$   are 

• 4,25,36

• 2,5,3

•  $\frac{1}{4},\frac{1}{25},\frac{1}{36}$  

•  $\frac{1}{2},\frac{1}{5},\frac{1}{6}$