Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 1 - Applications of Matrices and Determinants [Latest edition]

#### Chapters ## Chapter 1: Applications of Matrices and Determinants

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Exercise 1.6Exercise 1.7Exercise 1.8
Exercise 1.1 [Pages 15 - 16]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.1 [Pages 15 - 16]

Exercise 1.1 | Q 1. (i) | Page 15

Find the adjoint of the following:

[(-3, 4),(6,2)]

Exercise 1.1 | Q 1. (ii) | Page 15

Find the adjoint of the following:

[(2, 3, 1),(3, 4, 1),(3, 7, 2)]

Exercise 1.1 | Q 1. (iii) | Page 15

Find the adjoint of the following:1/3[(2, 2, 1),(-2, 1, 2),(1, -2, 2)]

Exercise 1.1 | Q 2. (i) | Page 15

Find the inverse (if it exists) of the following:

[(-2, 4),(1, -3)]

Exercise 1.1 | Q 2. (ii) | Page 15

Find the inverse (if it exists) of the following:

[(5, 1, 1),(1, 5, 1),(1, 1, 5)]

Exercise 1.1 | Q 2. (iii) | Page 15

Find the inverse (if it exists) of the following:

[(2, 3, 1),(3, 4, 1),(3, 7, 2)]

Exercise 1.1 | Q 3 | Page 15

If "F"(alpha) = [(cosalpha, 0, sinalpha),(0, 1, 0),(-sinalpha, 0, cosalpha)], show that ["F"(alpha)]^-1 = "F"(- alpha)

Exercise 1.1 | Q 4 | Page 15

If A = [(5, 3),(-1, -2)], show that A2 – 3A – 7I2 = O2. Hence find A–1

Exercise 1.1 | Q 5 | Page 15

If A = 1/9[(-8, 1, 4),(4, 4, 7),(1, -8, 4)], prove that "A"^-1 = "A"^"T"

Exercise 1.1 | Q 6 | Page 16

If A = [(8, -4),(-5, 3)], verify that A(adj A) = (adj A)A = |A|I2

Exercise 1.1 | Q 7 | Page 16

If A = [(3, 2),(7, 5)] and B = [(-1, -3),(5, 2)], verify that (AB)–1 = B1 A1

Exercise 1.1 | Q 8 | Page 16

If adj(A) = [(2, -4, 2),(-3, 12, -7),(-2, 0, 2)], find A

Exercise 1.1 | Q 9 | Page 16

If adj(A) = [(0, -2, 0),(6, 2, -6),(-3, 0, 6)], find A–1

Exercise 1.1 | Q 10 | Page 16

Find adj(adj(A)) if adj A = [(1, 0, 1),(0, 2, 0),(-1, 0, 1)]

Exercise 1.1 | Q 11 | Page 16

A = [(1, tanx),(-tanx, 1)], show that AT A–1 = [(cos 2x,  - sin 2x),(sin 2x, cos 2x)]

Exercise 1.1 | Q 12 | Page 16

Find the matrix A for which A[(5, 3),(-1, -2)] = [(14, 7),(7, 7)]

Exercise 1.1 | Q 13 | Page 16

Given A = [(1, -1),(2, 0)], B = [(3, -2),(1, 1)] and C = [(1, 1),(2, 2)], find a martix X such that AXB = C

Exercise 1.1 | Q 14 | Page 16

If A = [(0, 1, 1),(1, 0, 1),(1, 1, 0)], show that "A"^-1 = 1/2("A"^2 - 3"I")

Exercise 1.1 | Q 15 | Page 16

Decrypt the received encoded message [2 – 3][20 – 4] with the encryption matrix [(-1, -1),(2, 1)] and the decryption matrix as its inverse, where the system of codes are described by the numbers 1 – 26 to the letters A – Z respectively, and the number 0 to a blank space

Exercise 1.2 [Page 27]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.2 [Page 27]

Exercise 1.2 | Q 1. (i) | Page 27

Find the rank of the following matrices by minor method:

[(2, -4),(-1, 2)]

Exercise 1.2 | Q 1. (ii) | Page 27

Find the rank of the following matrices by minor method:

[(-1, 3),(4, 7),(3, 4)]

Exercise 1.2 | Q 1. (iii) | Page 27

Find the rank of the following matrices by minor method:

[(1, -2 - 1, 0),(3, -6 - 3, 1)]

Exercise 1.2 | Q 1. (iv) | Page 27

Find the rank of the following matrices by minor method:

[(1, -2, 3),(2, 4, -6),(5, 1, -1)]

Exercise 1.2 | Q 1. (v) | Page 27

Find the rank of the following matrices by minor method:

[(0, 1 ,2, 1),(0, 2, 4, 3),(8, 1, 0, 2)]

Exercise 1.2 | Q 2. (i) | Page 27

Find the rank of the following matrices by row reduction method:

[(1, 1, 1, 3),(2, -1, 3, 4),(5, -1, 7, 11)]

Exercise 1.2 | Q 2. (ii) | Page 27

Find the rank of the following matrices by row reduction method:

[(1, 2, -1),(3, -1, 2),(1, -2, 3),(1, -1, 1)]

Exercise 1.2 | Q 2. (iii) | Page 27

Find the rank of the following matrices by row reduction method:

[(3, -8, 5, 2),(2, -5, 1, 4),(-1, 2, 3, -2)]

Exercise 1.2 | Q 3. (i) | Page 27

Find the inverse of the following by Gauss-Jordan method:

[(2, -1),(5, -2)]

Exercise 1.2 | Q 3. (ii) | Page 27

Find the inverse of the following by Gauss-Jordan method:

[(1, -1, 0),(1, 0, -1),(6, -2, -3)]

Exercise 1.2 | Q 3. (iii) | Page 27

Find the inverse of the following by Gauss-Jordan method:

[(1, 2, 3),(2, 5, 3),(1, 0, 8)]

Exercise 1.3 [Page 33]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.3 [Page 33]

Exercise 1.3 | Q 1. (i) | Page 33

Solve the following system of linear equations by matrix inversion method:

2x + 5y = – 2, x + 2y = – 3

Exercise 1.3 | Q 1. (ii) | Page 33

Solve the following system of linear equations by matrix inversion method:

2x – y = 8, 3x + 2y = – 2

Exercise 1.3 | Q 1. (iii) | Page 33

Solve the following system of linear equations by matrix inversion method:

2x + 3y – z = 9, x + y + z = 9, 3x – y – z = – 1

Exercise 1.3 | Q 1. (iv) | Page 33

Solve the following system of linear equations by matrix inversion method:

x + y + z – 2 = 0, 6x – 4y + 5z – 31 = 0, 5x + 2y + 2z = 13

Exercise 1.3 | Q 2 | Page 33

If A = [(-5, 1, 3),(7, 1, -5),(1, -1, 1)] and B = [(1, 1, 2),(3, 2, 1),(2, 1, 3)], Find the products AB and BA and hence solve the system of equations x + y + 2z = 1, 3x + 2y + z = 7, 2x + y + 3z = 2

Exercise 1.3 | Q 3 | Page 33

A man is appointed in a job with a monthly salary of certain amount and a fixed amount of annual increment. If his salary was ₹ 19,800 per month at the end of the first month after 3 years of service and ₹ 23,400 per month at the end of the first month after 9 years of service, find his starting salary and his annual increment. (Use matrix inversion method to solve the problem.)

Exercise 1.3 | Q 4 | Page 33

Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method

Exercise 1.3 | Q 5 | Page 33

The prices of three commodities A, B and C are ₹ x, y and z per units respectively. A person P purchases 4 units of B and sells two units of A and 5 units of C. Person Q purchases 2 units of C and sells 3 units of A and one unit of B . Person R purchases one unit of A and sells 3 unit of B and one unit of C. In the process, P, Q and R earn ₹ 15,000, ₹ 1,000 and ₹ 4,000 respectively. Find the prices per unit of A, B and C. (Use matrix inversion method to solve the problem.)

Exercise 1.4 [Pages 35 - 36]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.4 [Pages 35 - 36]

Exercise 1.4 | Q 1. (i) | Page 35

Solve the following systems of linear equations by Cramer’s rule:

5x – 2y + 16 = 0, x + 3y – 7 = 0

Exercise 1.4 | Q 1. (ii) | Page 35

Solve the following systems of linear equations by Cramer’s rule:

3/2 + 2y = 12, 2/x + 3y = 13

Exercise 1.4 | Q 1. (iii) | Page 35

Solve the following systems of linear equations by Cramer’s rule:

3x + 3y – z = 11, 2x – y + 2z = 9, 4x + 3y + 2z = 25

Exercise 1.4 | Q 1. (iv) | Page 35

Solve the following systems of linear equations by Cramer’s rule:

3/x - 4/y - 2/z - 1 = 0, 1/x + 2/y + 1/z - 2 = 0, 2/x - 5/y - 4/z + 1 = 0

Exercise 1.4 | Q 2 | Page 35

In a competitive examination, one mark is awarded for every correct answer while 1/4 mark is deducted for every wrong answer. A student answered 100 questions and got 80 marks. How many questions did he answer correctly? (Use Cramer’s rule to solve the problem).

Exercise 1.4 | Q 3 | Page 34

A chemist has one solution which is 50% acid and another solution which is 25% acid. How much each should be mixed to make 10 litres of a 40% acid solution? (Use Cramer’s rule to solve the problem).

Exercise 1.4 | Q 4 | Page 35

A fish tank can be filled in 10 minutes using both pumps A and B simultaneously. However, pump B can pump water in or out at the same rate. If pump B is inadvertently run in reverse, then the tank will be filled in 30 minutes. How long would it take each pump to fill the tank by itself? (Use Cramer’s rule to solve the problem)

Exercise 1.4 | Q 5 | Page 36

A family of 3 people went out for dinner in a restaurant. The cost of two dosai, three idlies and two vadais is ₹ 150. The cost of the two dosai, two idlies and four vadais is ₹ 200. The cost of five dosai, four idlies and two vadais is ₹ 250. The family has ₹ 350 in hand and they ate 3 dosai and six idlies and six vadais. Will they be able to manage to pay the bill within the amount they had?

Exercise 1.5 [Page 37]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.5 [Page 37]

Exercise 1.5 | Q 1. (i) | Page 37

Solve the following systems of linear equations by Gaussian elimination method:

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

Exercise 1.5 | Q 1. (ii) | Page 37

Solve the following systems of linear equations by Gaussian elimination method:

2x + 4y + 6z = 22, 3x + 8y + 5z = 27, – x + y + 2z = 2

Exercise 1.5 | Q 2 | Page 37

If ax² + bx + c is divided by x + 3, x – 5, and x – 1, the remainders are 21, 61 and 9 respectively. Find a, b and c. (Use Gaussian elimination method.)

Exercise 1.5 | Q 3 | Page 37

An amount of ₹ 65,000 is invested in three bonds at the rates of 6%, 8% and 9% per annum respectively. The total annual income is ₹ 4,800. The income from the third bond is ₹ 600 more than that from the second bond. Determine the price of each bond. (Use Gaussian elimination method.)

Exercise 1.5 | Q 4 | Page 37

A boy is walking along the path y = ax2 + bx + c through the points (– 6, 8), (– 2, – 12), and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend? (Use Gaussian elimination method.)

Exercise 1.6 [Page 42]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.6 [Page 42]

Exercise 1.6 | Q 1. (i) | Page 42

Test for consistency and if possible, solve the following systems of equations by rank method:

x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4

Exercise 1.6 | Q 1. (ii) | Page 42

Test for consistency and if possible, solve the following systems of equations by rank method:

3x + y + z = 2, x – 3y + 2z = 1, 7x – y + 4z = 5

Exercise 1.6 | Q 1. (iii) | Page 42

Test for consistency and if possible, solve the following systems of equations by rank method:

2x + 2y + z = 5, x – y + z = 1, 3x + y + 2z = 4

Exercise 1.6 | Q 1. (iv) | Page 42

Test for consistency and if possible, solve the following systems of equations by rank method:

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

Exercise 1.6 | Q 2. (i) | Page 42

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have no solution

Exercise 1.6 | Q 2. (ii) | Page 42

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have unique solution

Exercise 1.6 | Q 2. (iii) | Page 42

Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have infinitely many solution

Exercise 1.6 | Q 3. (i) | Page 42

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have no solution

Exercise 1.6 | Q 3. (ii) | Page 42

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution

Exercise 1.6 | Q 3. (iii) | Page 42

Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have an infinite number of solutions

Exercise 1.7 [Page 47]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.7 [Page 47]

Exercise 1.7 | Q 1. (i) | Page 47

Solve the following system of homogenous equations:

3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0

Exercise 1.7 | Q 1. (ii) | Page 47

Solve the following system of homogenous equations:

2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0

Exercise 1.7 | Q 2. (i) | Page 47

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a unique solution

Exercise 1.7 | Q 2. (ii) | Page 47

Determine the values of λ for which the following system of equations x + y + 3z = 0; 4x + 3y + λz = 0, 2x + y + 2z = 0 has a non-trivial solution

Exercise 1.7 | Q 3 | Page 47

By using Gaussian elimination method, balance the chemical reaction equation:

$\ce{C2H + O2 -> H2O + CO2}$

Exercise 1.8 [Pages 48 - 50]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 1 Applications of Matrices and Determinants Exercise 1.8 [Pages 48 - 50]

Exercise 1.8 | Q 1 | Page 48

Choose the correct alternative:

If |adj(adj A)| = |A|9, then the order of the square matrix A is

• 3

• 4

• 2

• 5

Exercise 1.8 | Q 2 | Page 48

Choose the correct alternative:

If A is a 3 × 3 non-singular matrix such that AAT = AT A and B = A-1AT, then BBT =

• A

• B

•  I3

• BT

Exercise 1.8 | Q 3 | Page 48

Choose the correct alternative:

If A = [(3, 5),(1, 2)] B = adj A and C = 3A, then (|"adj B"|)/|"C"| =

• 1/3

• 1/9

• 1/4

• 1

Exercise 1.8 | Q 4 | Page 48

Choose the correct alternative:

If A = [(1, -2),(1, 4)] = [(6, 0),(0, 6)], then A =

• [(1, -2),(1, 4)]

• [(1, 2),(-1, 4)]

• [(4, 2),(-1, 1)]

• [(4, -1),(2, 1)]

Exercise 1.8 | Q 5 | Page 48

Choose the correct alternative:

If A = [(7, 3),(4, 2)] then 9I2 – A =

• "A"^-1

• "A"^-1/2

• 3"A"^-1

• 2"A"^-1

Exercise 1.8 | Q 6 | Page 48

Choose the correct alternative:

If A = [(2, 0),(1, 5)] and B = [(1, 4),(2, 0)] then |adj (AB)| =

• – 40

• – 80

• – 60

• – 20

Exercise 1.8 | Q 7 | Page 48

Choose the correct alternative:

If + = [(1, x, 0),(1, 3, 0),(2, 4, -2)] is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

• 15

• 12

• 14

• 11

Exercise 1.8 | Q 8 | Page 48

Choose the correct alternative:

If A = [(3, 1, -1),(2, -2, 0),(1, 2, -1)] and "A"^-1 = [("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)] then the value of a23 is

• 0

• – 2

• – 3

• – 1

Exercise 1.8 | Q 9 | Page 48

Choose the correct alternative:

If A B, and C are invertible matrices of some order, then which one of the following is not true?

• det A–1 = (det A)1

• (ABC)1 = C1B1A1

Exercise 1.8 | Q 10 | Page 48

Choose the correct alternative:

If ("AB")^-1 = [(12, -17),(-19, 27)] and "A"^-1 = [(1, -1),(-2, 3)] then "B"^-1 =

• [(2, -5),(- 3, 8)]

• [(8, 5),(3, 2)]

• [(3, 1),(2, 1)]

• [(8, -5),(-3, 2)]

Exercise 1.8 | Q 11 | Page 49

Choose the correct alternative:

If ATA1 is symmetric, then A2 =

• A–1

• (AT)2

• AT

• (A1)2

Exercise 1.8 | Q 12 | Page 49

Choose the correct alternative:

If A is a non-singular matrix such that A–1 = [(5, 3),(-2, -1)], then (AT)1 =

• [(-5, 3),(2, 1)]

• [(5, 3),(-2, -1)]

• [(-1, -3),(2, 5)]

• [(5, -2),(3, -1)]

Exercise 1.8 | Q 13 | Page 49

Choose the correct alternative:

If A = [(3/5, 4/5),(x, 3/5)] and AT = A–1, then the value of x is

• (-4)/5

• (-3)/5

• 3/5

• 4/5

Exercise 1.8 | Q 14 | Page 49

Choose the correct alternative:

If A = [(1, tan  theta/2),(- tan theta/2, 1)] and AB = I2, then B =

• (cos^2 theta/2)"A"

• (cos^2 theta/2)"A"^"T"

• (cos^2theta)"I"

• (sin^2  theta/2)"A"

Exercise 1.8 | Q 15 | Page 49

Choose the correct alternative:

If A = [(costheta, sintheta),(-sintheta, costheta)] and A(adj A) = [("k", 0),(0, "k")], then k =

• 0

• sin θ

• cos θ

• 1

Exercise 1.8 | Q 16 | Page 49

Choose the correct alternative:

If A = [(2, 3),(5, -2)] be such that λA–1 = A, then λ is

• 17

• 14

• 19

• 21

Exercise 1.8 | Q 17 | Page 49

Choose the correct alternative:

If adj A = [(2, 3),(4, 1)] and adj B = [(1, -2),(-3, 1)] then adj (AB) is

• [(-7, -1),(7, -9)]

• [(-6, 5),(-2, -10)]

• [(-7, 7),(-1, -9)]

• [(-6, -2),(5, -10)]

Exercise 1.8 | Q 18 | Page 49

Choose the correct alternative:

The rank of the matrix [(1, 2, 3, 4),(2, 4, 6, 8),(-1, -2, -3, -4)] is

• 1

• 2

• 4

• 3

Exercise 1.8 | Q 19 | Page 49

Choose the correct alternative:

If xayb = em, xcyd = en, Δ1 = |("m", "b"),("n", "d")|, Δ2 = |("a", "m"),("c", "n")|, Δ3 = |("a", "b"),("c", "d")|, then the values of x and y are respectively,

• "e"^(Delta_2/Delta_1), "e"^(Delta_3/Delta_1)

• log(Delta_1/Delta_3), log(Delta_2/Delta_3)

• log(Delta_2/Delta_1), log(Delta_3/Delta_1)

• "e"^(Delta_1/Delta_3), "e"^(Delta_2/Delta_3)

Exercise 1.8 | Q 20 | Page 50

Choose the correct alternative:

Which of the following is/are correct?
(i) Adjoint of a symmetric matrix is also a symmetric matrix.
(ii) Adjoint of a diagonal matrix is also a diagonal matrix.
(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj (A).

• Only (i)

• (ii) and (iii)

• (iii) and (iv)

• (i), (ii) and (iv)

Exercise 1.8 | Q 21 | Page 50

Choose the correct alternative:

If ρ(A) ρ([A|B]), then the system AX = B of linear equations is

• consistent and has a unique solution

• consistent

• consistent and has infinitely many solution

• inconsistent

Exercise 1.8 | Q 22 | Page 50

Choose the correct alternative:

If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ) x – y + z = 0, (sin θ) x + y + z = 0 has a non-trivial solution then θ is

• (2pi)/3

• (3pi)/4

• (5pi)/6

• pi/4

Exercise 1.8 | Q 23 | Page 50

Choose the correct alternative:

The augmented matrix of a system of linear equations is [(1, 2, 7, 3),(0, 1, 4, 6),(0, 0, lambda - 7, mu + 7)]. This system has infinitely many solutions if

• λ = 7, µ ≠ – 5

• λ = – 7, µ = 5

• λ ≠ 7, µ ≠ – 5

• λ = 7, µ = – 5

Exercise 1.8 | Q 24 | Page 50

Choose the correct alternative:

Let A = [(2, -1, 1),(-1, 2, -1),(1, -1, 2)] and 4B = [(3, 1, -1),(1, 3, x),(-1, 1, 3)]. If B is the inverse of A, then the value of x is

• 2

• 4

• 3

• 1

Exercise 1.8 | Q 25 | Page 50

Choose the correct alternative:

If A = [(3, -3, 4),(2, -3, 4),(0, -1, 1)], then adj(adj A) is

• [(3, -3, 4),(2, -3, 4),(0, -1, 1)]

• [(6, -6, 8),(4, -6, 8),(0, -2, 2)]

• [(-3, 3, -4),(-2, 3, -4),(0, 1, -1)]

• [(3, -3, 4),(0, -1, 1),(2, -3, 4)]

## Chapter 1: Applications of Matrices and Determinants

Exercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Exercise 1.6Exercise 1.7Exercise 1.8 ## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 1 - Applications of Matrices and Determinants

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 1 Applications of Matrices and Determinants are Introduction to Applications of Matrices and Determinants, Inverse of a Non-singular Square Matrix, Elementary Transformations of a Matrix, Applications of Matrices: Solving System of Linear Equations, Applications of Matrices: Consistency of System of Linear Equations by Rank Method.

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