Find the rank of the matrix \(\left[ \begin{matrix} 7 & -1 \\ 2 & 1 \end{matrix} \right] \)

Find the rank of the matrix \(\left( \begin{matrix} 2 & -4 \\ -1 & 2 \end{matrix} \right) \)

Solve x + 2y = 3 and x +y = 2 using Cramer's rule.

Solve: x + 2y = 3 and 2x + 4y = 6 using rank method.

Show that the equations x + y + z = 6, x + 2y + 3z = 14 and x + 4y + 7z = 30 are consistent

Solve: 2x + 3y = 4 and 4x + 6y = 8 using Cramer's rule.

If A and B are non-singular matrices, prove that AB is non-singular.

For what value of x, the matrix
\(A=\left| \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right| \) is singular?

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

Two newspapers A and B are published in a city . Their market shares are 15% for A and 85% for B of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year

Find the rank of \(\left(\begin{array}{cc} 7 & -1 \\ 2 & 1 \end{array}\right)\)

Find the rank of the matrix \(\left(\begin{array}{ccc} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{array}\right)\)

If \(\mathrm{A}=\left(\begin{array}{ccc} x & x & x \\ 4 & -2 & 1 \\ 2 & 3 & 4 \end{array}\right)\)find x if \(\rho(\mathrm{A})=3\)

Find the rank of the matrix \(\left(\begin{array}{rrrr} 1 & 2 & 3 & -1 \\ 2 & 4 & 6 & -2 \\ 3 & -6 & 9 & -3 \end{array}\right)\)

Find the rank of the following matrices.
(i) \(\left(\begin{array}{ccc} 2 & 4 & 5 \\ 4 & 8 & 10 \\ -6 & -12 & -15 \end{array}\right)\)
(ii) \(\left(\begin{array}{cccc} 1 & -3 & 4 & 7 \\ 9 & 1 & 2 & 0 \end{array}\right)\)
(iii) \(\left(\begin{array}{lll} 3 & 2 & 1 \\ 0 & 4 & 5 \\ 3 & 6 & 6 \end{array}\right)\)
(iv) \(\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{array}\right)\)
(v) \(\left(\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ -1 & -2 & -2 & -4 \end{array}\right)\)
(vi) \(\left(\begin{array}{cccc} 1 & 3 & 4 & 3 \\ 3 & 9 & 12 & 9 \\ 1 & 3 & 4 & 3 \end{array}\right)\)
(vii) \(\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right)\)
(viii) \(\left(\begin{array}{cc} 9 & 6 \\ -6 & 4 \end{array}\right)\)