For the matrix 

$A=\begin{bmatrix}1&1&2\\1&2&3\\0&-1&-1\end{bmatrix}$

For non singular matrices P and Q such that PAQ is in the normal form. Hence find the rank of A.

Reduce to triangular form

$\begin{bmatrix}3&-4&-5\\-9&1&4\\-5&3&1\end{bmatrix}$

Use Gauss-jordan method to find the inverse of matrix$\begin{bmatrix}8&4&3\\2&1&1\\1&2&1\end{bmatrix}$

Solve thehelp of matrix the simultaneous equations :

$x+y+z=3, x+2y+3z=4, x+4y+9z=6$

Solve the system of equations

$2x_{1}+x_{2}+2x_{3}+x_{4}=6$

$4x_{1}+3x_{2}+3x_{3}-3x_{4}=-1$

$6x_{1}-6x_{2}+6x_{3}+12x_{4}=36$

$2x_{1}+2x_{2}-x_{3}+x_{4}=10$

Using matrix mathod , show that the equations

3x+3y+2z=1,

x+2y=4,

10y+3z=-2,

2x-3y-z=5

are consistent and hence obtain the solution for x,y,and z.

For what value of the parameter $\lambda ,\mu$do the system of equations

1) no solution 

2) unique solution

3) more than one solution

$x+y+z=6,x+2y+3z=10,x+2y+\lambda{z}=\mu$

A, B and C can do a piece of work in 20, 30 and 60 days respectively. 
In how many days can A do the work if he is assisted by B and C on every third day?

A : 12days
B : 15days
C : 16days
D : 18days

Find the value of integration:

$\int{(3X+5)^{3}}dx$

Find the value of integration:

$\int\frac{2x+4}{5x+3}dx$

Find the root of equation:

$3x^2+2x+4$