## EXERCISE 4.6

#### Page No 136:

#### Question 1:

Examine the consistency of the system of equations.

*x *+ 2*y *= 2

2*x* + 3*y *= 3

#### Answer:

The given system of equations is:

*x *+ 2*y *= 2

2*x* + 3*y *= 3

The given system of equations can be written in the form of *AX* = *B*, where

∴ *A* is non-singular.

Therefore, *A*^{−1} exists.

Hence, the given system of equations is consistent.

#### Question 2:

Examine the consistency of the system of equations.

2*x *− *y* = 5

*x* + *y *= 4

#### Answer:

The given system of equations is:

2*x *− *y* = 5

*x* + *y *= 4

The given system of equations can be written in the form of *AX* = *B*, where

∴ *A* is non-singular.

Therefore, *A*^{−1} exists.

Hence, the given system of equations is consistent.

#### Question 3:

Examine the consistency of the system of equations.

*x* + 3*y* = 5

2*x* + 6*y* = 8

#### Answer:

The given system of equations is:

*x* + 3*y* = 5

2*x* + 6*y* = 8

The given system of equations can be written in the form of *AX* = *B*, where

∴ *A* is a singular matrix.

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

#### Question 4:

Examine the consistency of the system of equations.

*x* +* y *+ *z* = 1

2*x* + 3*y* + 2*z* = 2

*ax* + *ay* + 2*az* = 4

#### Answer:

The given system of equations is:

*x* +* y *+ *z* = 1

2*x* + 3*y* + 2*z* = 2

*ax* + *ay* + 2*az* = 4

This system of equations can be written in the form *AX* = *B*, where

∴ *A* is non-singular.

Therefore, *A*^{−1} exists.

Hence, the given system of equations is consistent.

#### Question 5:

Examine the consistency of the system of equations.

3*x* −* y *− 2z = 2

2*y* − *z* = −1

3*x* − 5*y* = 3

#### Answer:

The given system of equations is:

3*x* −* y *− 2z = 2

2*y* − *z* = −1

3*x* − 5*y* = 3

This system of equations can be written in the form of *AX* = *B*, where

∴ *A* is a singular matrix.

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

#### Question 6:

Examine the consistency of the system of equations.

5*x* −* y *+ 4*z* = 5

2*x* + 3*y* + 5*z* = 2

5*x* − 2*y* + 6*z* = −1

#### Answer:

The given system of equations is:

5*x* −* y *+ 4*z* = 5

2*x* + 3*y* + 5*z* = 2

5*x* − 2*y* + 6*z* = −1

This system of equations can be written in the form of *AX* = *B*, where

∴ *A* is non-singular.

Therefore, *A*^{−1} exists.

Hence, the given system of equations is consistent.

#### Question 7:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 8:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 9:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 10:

Solve system of linear equations, using matrix method.

5*x* + 2*y* = 3

3*x* + 2*y* = 5

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 11:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 12:

Solve system of linear equations, using matrix method.

*x* − *y* + *z* = 4

2*x* + *y* − 3*z* = 0

*x* + *y* + *z* = 2

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 13:

Solve system of linear equations, using matrix method.

2*x* + 3*y* + 3*z* = 5

*x* − 2*y* + *z* = −4

3*x* − *y* − 2*z* = 3

#### Answer:

The given system of equations can be written in the form *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Question 14:

Solve system of linear equations, using matrix method.

*x* − *y* + 2*z* = 7

3*x* + 4*y* − 5*z* = −5

2*x* −* y* + 3*z* = 12

#### Answer:

The given system of equations can be written in the form of *AX* = *B*, where

Thus, *A* is non-singular. Therefore, its inverse exists.

#### Page No 137:

#### Question 15:

If, find *A*^{−1}. Using A^{−1} solve the system of equations

#### Answer:

Now, the given system of equations can be written in the form of *AX* = *B*, where

#### Question 16:

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg

wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70.

Find cost of each item per kg by matrix method.

#### Answer:

Let the cost of onions, wheat, and rice per kg be Rs *x*, Rs *y*,and Rs *z* respectively.

Then, the given situation can be represented by a system of equations as:

This system of equations can be written in the form of *AX* = *B*, where

Now,

*X* = *A*^{−1} *B*

Hence, the cost of onions is Rs 5 per kg, the cost of wheat is Rs 8 per kg, and the cost of rice is Rs 8 per kg.