Find the rank of the matrix \(\left( \begin{matrix} 0 & -1 & 5 \\ 2 & 4 & -6 \\ 1 & 1 & 5 \end{matrix} \right) \)

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

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

Find the rank of the matrix A = \(\left( \begin{matrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{matrix}\begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \)

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

Show that the equations x + y = 5, 2x + y = 8 are consistent and solve them.

Show that the equations 2x + y = 5,4x + 2y = 10 are consistent and solve them.

Show that the equations 3x − 2y = 6, 6x − 4y = 10 are inconsistent

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.

Show that the following system of equations have unique solution:
x + y + z = 3, x + 2y + 3z = 4, x + 4y + 9z = 6 by rank method.

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

The following table represents the number of shares of two companies A and B during the month of January and February and it also gives the amount in rupees invested by Ravi during these two months for the purchase of shares of two companies. Find the the price per share of A and B purchased during both the months

Months	Number of Shares of the company		Amount invested by Ravi (in Rs)
	A	B
January	10	5	125
February	9	12	150

The total cost of 11 pencils and 3 erasers is Rs. 64 and the total cost of 8 pencils and 3 erasers is Rs. 49. Find the cost of each pencil and each eraser by Cramer’s rule.

Solve the following equations by using Cramer’s rule
 2x + 3y = 7; 3x + 5y = 9

A commodity was produced by using 3 units of labour and 2 units of capital, the total cost is Rs 62. If the commodity had been produced by using 4 units of labour and one unit of capital, the cost is Rs 56. What is the cost per unit of labour and capital? (Use determinant method).

A total of Rs. 8,600 was invested in two accounts. One account earned \(4\frac { 3 }{ 4 } %\)% annual interest and the other earned \(6\frac { 1 }{ 2 } %\)% annual interest. If the total interest for one year was Rs. 431.25, how much was invested in each account? (Use determinant method).

At marina two types of games viz., Horse riding and Quad Bikes riding are available on hourly rent. Keren and Benita spent Rs. 780 and Rs. 560 during the month of May.

Name	Number of hours		Total amount spent (in Rs)
	Horse Riding	Quad Bike Riding
Keren	3	4	780
Benita	2	3	560

Find the hourly charges for the two games (rides). (Use determinant method).

Consider the matrix of transition probabilities of a product available in the market in two brands A and B.
\(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.9 } & \overset { B }{ 0.1 } \\ 0.3 & 0.7 \end{matrix} \right) \)
Determine the market share of each brand in equilibrium position.

Parithi is either sad (S) or happy (H) each day. If he is happy in one day, he is sad on the next day by four times out of five. If he is sad on one day, he is happy on the next day by two times out of three. Over a long run, what are the chances that Parithi is happy on any given day?

Akash bats according to the following traits. If he makes a hit (S), there is a 25% chance that he will make a hit his next time at bat. If he fails to hit (F), there is a 35% chance that he will make a hit his next time at bat. Find the transition probability matrix for the data and determine Akash’s long- range batting average.

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 45% of those who already subscribe will subscribe again while 30% of those who do not now subscribe will subscribe. On the last letter, it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

Find the rank of the matrix A =\(\left( \begin{matrix} -2 & 1 & 3 \\ 0 & 1 & 1 \\ 1 & 3 & 4 \end{matrix}\begin{matrix} 4 \\ 2 \\ 7 \end{matrix} \right) \)

Find the rank of the matrix A =\(\left( \begin{matrix} 4 & 5 & 2 \\ 3 & 2 & 1 \\ 4 & 4 & 8 \end{matrix}\begin{matrix} 2 \\ 6 \\ 0 \end{matrix} \right) \)

Examine the consistency of the system of equations: x + y + z = 7, x + 2y + 3z = 18, y + 2z = 6.

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current letter can be expected to order a subscription?

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 2 & -1 & 1 \\ 3 & 1 & -5 \\ 1 & 1 & 1 \end{matrix} \right) \)

Find the rank of each of the following matrices.
\(\left( \begin{matrix} -1 & 2 & -2 \\ 4 & -3 & 4 \\ -2 & 4 & -4 \end{matrix} \right) \)

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

Find the rank of each of the following matrices.
\(\left( \begin{matrix} 3 & 1 & -5 \\ 1 & -2 & 1 \\ 1 & 5 & -7 \end{matrix}\begin{matrix} -1 \\ -5 \\ 2 \end{matrix} \right) \)

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

Solve the following equation by using Cramer’s rule
5x + 3y = 17; 3x + 7y = 31