Find the rank of the matrix
\(A=\left( \begin{matrix} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \end{matrix} \right) \)

Find the rank of the matrix \(A=\left( \begin{matrix} 1 & 2 & -4 \\ 2 & -1 & 3 \\ 8 & 1 & 9 \end{matrix}\begin{matrix} 5 \\ 6 \\ 7 \end{matrix} \right) \)

Show that the equations 2x - y + z = 7, 3x + y - 5z = 13, x + y + z = 5 are consistent and have a unique solution.

Show that the equations x + 2y = 3, y - z = 2, x + y + z = 1 are consistent and have infinite sets of solution.

Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

Solve: 2x - 3y - 1 = 0, 5x + 2y - 12 = 0 by Cramer's rule.

If \(\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right] \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] =\left[ \begin{matrix} 2 \\ -1 \\ 3 \end{matrix} \right] \) find x,y and z

If \(A=\left( \begin{matrix} 2 & 4 \\ 4 & 3 \end{matrix} \right) ,X=\left( \begin{matrix} n \\ 1 \end{matrix} \right) B=\left( \begin{matrix} 8 \\ 11 \end{matrix} \right) \) and AX = B then find n.

Solve: 2x + 3y = 5, 6x + 5y = 11

Two products A and B currently share the market with shares 60% and 40% each respectively. Each week some brand switching latees place. Of those who bought A the previous week 70% buy it again whereas 30% switch over to B. Of those who bought B the previous week, 80% buy it again whereas 20% switch over to A. Find their shares after one week and after two weeks.

Find the rank of the following matrices 
(i) \(\left(\begin{array}{cccc} 1 & 1 & 1 & 3 \\ 2 & -1 & 3 & 4 \\ 5 & -1 & 7 & 11 \end{array}\right)\)
(ii) \(\left(\begin{array}{llll} 4 & 2 & 1 & 3 \\ 6 & 3 & 4 & 7 \\ 2 & 1 & 0 & 1 \end{array}\right)\)
(iii) \(\left(\begin{array}{cccc} 3 & 1 & -5 & -1 \\ 1 & -2 & 1 & -5 \\ 1 & 5 & -7 & 2 \end{array}\right)\)
(iv) \(\left(\begin{array}{cccc} 3 & 1 & 2 & 0 \\ 1 & 0 & -1 & 0 \\ 2 & 1 & 3 & 0 \end{array}\right)\)
(v) \(\left(\begin{array}{cccc} 0 & 1 & 2 & 1 \\ 2 & -3 & 0 & -1 \\ 1 & -1 & -1 & 0 \end{array}\right)\)
(vi) \(\left(\begin{array}{cccc} 1 & 2 & -1 & 3 \\ 2 & 4 & 1 & -2 \\ 3 & 6 & 3 & -7 \end{array}\right)\)
(vii) \(\left(\begin{array}{cccc} 1 & -2 & 3 & 4 \\ -2 & 4 & -1 & -3 \\ -1 & 2 & 7 & 6 \end{array}\right)\)

Find the ranks of A + B and AB where \(A=\left(\begin{array}{ccc} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{array}\right)\) and \(B=\left(\begin{array}{ccc} -1 & -2 & -1 \\ 6 & 12 & 6 \\ 5 & 10 & 5 \end{array}\right)\)

Show that the equations \(x-3 y-8 z=-10\), \(3 x+y-4 z=0,2 x+5 y+6 z=13\) are consistent and have infinite sets of solution.

Test the system of equations \(4 x-5 y-2 z=2\), \(5 x-4 y+2 z=-2,2 x+2 y+8 z=-1\) for consistency.

Show that the equations \(x+y+z=-3,3 x+y\)\(-2 z=-2,2 x+4 y+7 z=7\) are not consistent.