Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 1 - Applications of Matrices and Determinants [Latest edition]

Find the rank of the following matrices

`((5, 6),(7, 8))`

Find the rank of the following matrices

`((1, -1),(3, -6))`

Find the rank of the following matrices

`((1, 4),(2, 8))`

Find the rank of the following matrices

`((2, -1, 1),(3, 1, -5),(1, 1, 1))`

Find the rank of the following matrices

`((-1, 2, -2),(4, -3, 4),(-2, 4, -4))`

Find the rank of the following matrices

`((1, 2, -1, 3),(2, 4, 1, -2),(3, 6, 3, -7))`

Find the rank of the following matrices

`((3, 1, -5, -1),(1, -2, 1, -5),(1, 5, -7, 2))`

Find the rank of the following matrices

`((1, -2, 3, 4),(-2, 4, -1, -3),(-1, 2, 7, 6))`

If A = `((1, 1, -1),(2, -3, 4),(3, -2, 3))` and B = `((1, -2, 3),(-2, 4, -6),(5, 1, -1))`, then find the rank of AB and the rank of BA.

Solve the following system of equations by rank method

x + y + z = 9, 2x + 5y + 7z = 52, 2x – y – z = 0

Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them by rank method

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

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

The price of three commodities, X, Y and Z are and z respectively Mr. Anand purchases 6 units of Z and sells 2 units of X and 3 units of Y. Mr.Amar purchases a unit of Y and sells 3 units of X and 2 units of Z. Mr. Amit purchases a unit of X and sells 3 units of Y and a unit of Z. In the process they earn ₹ 5,000/-, ₹ 2,000/- and ₹ 5,500/- respectively. Find the prices per unit of three commodities by rank method

An amount of ₹ 5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is ₹ 358/-. If the income from the first two investments is ₹ 70/- more than the income from the third, then find the amount of investment in each bond by the rank method

Solve the following equations by using Cramer’s rule:

2x + 3y = 7; 3x + 5y = 9

Solve the following equations by using Cramer’s rule:

5x + 3v = 17; 3x + 7y = 31

Solve the following equations by using Cramer’s rule:

2x + y – z = 3, x + y + z – 1, x – 2y – 3z = 4

Solve the following equations by using Cramer’s rule:

x + y + z = 6, 2x + 3y – z = 5, 6x – 2y – 3z = – 7

Solve the following equations by using Cramer’s rule:

x + 4y + 3z = 2, 2x – 6y + 6z = – 3, 5x – 2y + 3z = – 5

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

A total of ₹ 8,600 was invested in two accounts. One account earned `4 3/4`% annual interest and the other earned `6 1/2`% annual interest. If the total interest for one year was ₹ 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 ₹ 780 and ₹ 560 during the month of May.

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

In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity.

Find the weights assigned to the three varieties by using Cramer’s Rule

A total of ₹ 8,500 was invested in three interest-earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was ₹ 380 and the amount invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule)

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?

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train next year.

What percent of commuters will be using the transit system year after the next year?

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train next year.

What percent of commuters will be using the transit system in the long run?

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?

Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again where as 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?

Choose the correct alternative:

A = (1, 2, 3), then the rank of AA^T is

Choose the correct alternative:

The rank of m n × matrix whose elements are unity is

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If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.4&0.6\\0.2&0.8\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then at equilibriuium A is equal to

Choose the correct alternative:

If A = `((2, 0),(0, 8))`, then p(A) is

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The rank of the matrix `((1, 1, 1),(1, 2, 3),(1, 4, 9))` is

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The rank of the unit matrix of order n is

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

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If A = `((1),(2),(3))` then the rank of AA^T is

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If the rank of the matrix `[(lambda, -1, 0),(0, lambda, -1),(-1, 0, lambda)]` is 2, then λ is

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The rank of the diagonal matrix `[(1, , , , ,),(, 2, , , ,),(, , -3, , ,),(, , , 0, ,),(, , , , 0,),(, ,  , , ,0)]`

Choose the correct alternative:

If \[{\begin{matrix} & \begin{matrix}A&&B\end{matrix} \\ T = \begin{matrix}A\\B\end{matrix} & \begin{pmatrix}0.7&0.3\\0.6&x\end{pmatrix}\\ \end{matrix}}\] is a transition probability matrix, then the value of x is

Choose the correct alternative:

Which of the following is not an elementary transformation?

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If p(A) = p(A,B)= then the system is

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If p(A) = p(A, B)= the number of unknowns, then the system is

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If p(A) ≠ p(A, B) =, then the system is

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In a transition probability matrix, all the entries are greater than or equal to

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If the number of variables in a non-homogeneous system AX = B is n, then the system possesses a unique solution only when

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The system of equations 4x + 6y = 5, 6x + 9y = 7 has

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For the system of equations x + 2y + 3z = 1, 2x + y + 3z = 3, 5x + 5y + 9z = 4

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If |A| ≠ 0, then A is

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The system of linear equations x = y + z = 2, 2x + y – z = 3, 3x + 2y + k = 4 has unique solution, if k is not equal to

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Cramer’s rule is applicable only to get an unique solution when

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If `"a"_1/x + "b"_1/y = "c"_1, "a"_2/x + "b"_2/y = "c"_2, Delta_1 = |("a"_1, "b"_1),("a"_2, "b"_2)|, Delta_2 = |("b"_1, "c"_1),("b"_2, "c"_2)| Delta_3 = |("c"_1, "a"_1),("c"_2, "a"_2)|` then (x , y) is

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If `|"A"_("n" xx "n")|` = 3 and |adj A| = 243 then the value n is

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Rank of a null matrix is

Find the rank of the matrix

A = `((1, -3, 4, 7),(9, 1, 2, 0))`

Find the rank of the matrix

A = `((-2, 1, 3, 4),(0, 1, 1, 2),(1, 3, 4, 7))`

Find the rank of the matrix

A = `((4, 5, 2, 2),(3, 2, 1, 6),(4, 4, 8, 0))`

Examine the consistency of the system of equations:

x + y + z = 7, x + 2y + 3z = 18, y + 2z = 6

Find k if the equations 2x + 3y – z = 5, 3x – y + 4z = 2, x + 7y – 6z = k are consistent

Find k if the equations x + y + z = 1, 3x – y – z = 4, x + 5y + 5z = k are inconsistent

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

The cost of 2kg. of wheat and 1kg. of sugar is ₹ 100. The cost of 1kg. of wheat and 1kg. of rice is ₹ 80. The cost of 3kg. of wheat, 2kg. of sugar and 1kg of rice is ₹ 220. Find the cost of each per kg., using Cramer’s rule

A salesman has the following record of sales during three months for three items A, B and C, which have different rates of commission.

Find out the rate of commission on the items A, B and C by using Cramer’s rule

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?