The rank of the matrix  \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is ________.

If \(\rho (A)\) = r  then which of the following is correct?

If  \(T=\begin{array}{l} A \\ B \end{array}\left(\begin{array}{ll} 0.7 & 0.3 \\ 0.6 & x \end{array}\right)\) is a transition probability matrix, then the value of x is ________.

The rank of an n x n matrix each of whose elements is 2 is __________

If A, B are two n x n non-singular matrices, then ___________

If A=\(\left( \begin{matrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{matrix} \right) \) and B=\(\left( \begin{matrix} 1 & -2 & 3 \\ -2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right) \), then find the rank of AB and the rank of BA.

For what values of the parameter λ, will the following equations fail to have unique solution: 3x − y+λz = 1, 2x + y + z = 2, x + 2y − λz = −1 by rank method.

Two types of soaps A and B are in the market. Their present 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 and when is the equilibrium reached?

Solve the equations x + 2y + z = 7, 2x − y + 2z = 4, x + y − 2z = −1 by using Cramer’s rule

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

Find the rank of the matrix \(\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right) \)

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

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 = 5, 6x + 5y = 11

Solve the equations 2x + 3y = 7, 3x + 5y = 9 by Cramer’s rule.

Using determinants, find the quadratic defined by f(x) = ax² + bx + c if
f(1) = 0,
f(2) = - 2 and
f(3) = -6.