Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Matrices Solutions Exercise 3(g) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Matrices Solutions Exercise 3(g)

Examine whether the following systems of equations are consistent or inconsistent and if consistent find the complete solutions.

Question 1.

x + y + z = 4

2x + 5y – 2z = 3

x + 7y – 7z = 5

Solution:

ρ(A) = 2, ρ(AB) = 3

ρ(A) ≠ ρ(AB)

∴ The given system of equations are in consistent.

Question 2.

x + y + z = 6

x – y + z = 2

2x – y + 3z = 9

Solution:

Question 3.

x + y + z = 1

2x + y + z = 2

x + 2y + 2z = 1

Solution:

ρ(A) = 2 = ρ(AB) < 3

The given system of equations are consistent and have infinitely many solutions.

The solutions are given by [(x, y, z) 1x = 1, y + z = 0].

Question 4.

x + y + z = 9

2x + 5y + 7z = 52

2x + y – z = 0

Solution:

∴ ρ(A) = ρ(AB) = 3

The given system of equations are consistent have a unique solution.

∴ Solution is given by x = 1, y = 3, z = 5.

Question 5.

x + y + z = 6

x + 2y + 3z = 10

x + 2y + 4z = 1

Solution:

Augmented matrix A = \(\left[\begin{array}{cccc}

1 & 1 & 1 & 6 \\

1 & 2 & 3 & 10 \\

1 & 2 & 4 & 1

\end{array}\right]\)

By R_{2} → R_{2} – R_{1}, R_{3} → R_{3} – R_{2}, we obtain

∴ ρ(A) = ρ(AB) = 3

The given system of equations are consistent.

They have a unique solution.

∴ Solution is given by x = -7, y = 22, z = -9.

Question 6.

x – 3y – 8z = -10

3x + y – 4z = 0

2x + 5y + 6z = 13

Solution:

The Augmented matrix

ρ(A) = ρ(AB) = 2 < 3

∴ The given system of equations are consistent have infinitely many solutions.

x + y = 2 and y + 2z = 3

Taking z = k, y = 3 – 2z = 3-2k

x = 2 – y

= 2 – (3 – 2k)

= 2 – 3 + 2k

= 2k – 1

∴ The solutions are given by x = -1 + 2k, y = 3 – 2k, z = k where ‘k’ is any scalar.

Question 7.

2x + 3y + z = 9

x + 2y + 3z = 6

3x + y + 2z = 8

Solution:

∴ ρ(A) = ρ(AB) = 3

The given system of equations are consistent have a unique solution.

∴ Solution is given by x = \(\frac{35}{18}\), y = \(\frac{29}{18}\), z = \(\frac{5}{18}\)

Question 8.

x + y + 4z = 6

3x + 2y – 2z = 9

5x + y + 2z = 13

Solution:

∴ ρ(A) = ρ(AB) = 3

∴ The given system of equations are consistent have a unique solution.

∴ Solution is given by x = 2, y = 2, z = \(\frac{1}{2}\)