**Solve the following systems of homogeneous linear equations by matrix method:**

**Question 1. **

**2x – y + z = 0**

**3x + 2y – z = 0**

**x + 4y + 3z = 0**

**Solution:**

Given

2x – y + z = 0

3x + 2y – z = 0

X + 4y + 3z = 0

The system can be written as

A X = 0

Now, |A| = 2(6 + 4) + 1(9 + 1) + 1(12 – 2)

|A| = 2(10) + 10 + 10

|A| = 40 ≠ 0

Since, |A|≠ 0, hence x = y = z = 0 is the only solution of this homogeneous equation.

**Question 2. **

**2x – y + 2z = 0**

**5x + 3y – z = 0**

**X + 5y – 5z = 0**

**Solution:**

Given 2x – y + 2z = 0

5x + 3y – z = 0

X + 5y – 5z = 0

A X = 0

Now, |A| = 2(– 15 + 5) + 1(– 25 + 1) + 2(25 – 3)

|A| = – 20 – 24 + 44

|A| = 0

Thus, the system has infinite solutions

Let z = k

2x – y = – 2k

5x + 3y = k

**Question 3.**

**3x – y + 2z = 0**

**4x + 3y + 3z = 0**

**5x + 7y + 4z = 0**

**Solution:**

Given:

3x – y + 2z = 0

4x + 3y + 3z = 0

5x + 7y + 4z = 0

A X = 0

Now, |A| = 3(12 – 21) + 1(16 – 15) + 2(28 – 15)

|A| = – 27 + 1 + 26

|A| = 0

Hence, the system has infinite solutions

Let z = k

3x – y = – 2k

4x + 3y = – 3k

**Question 4. **

**x + y – 6z = 0**

**x – y + 2z = 0**

**– 3x + y + 2z = 0**

**Solution:**

Given:

x + y – 6z = 0

x – y + 2z = 0

– 3x + y + 2z = 0

A X = 0

Now, |A| = 1(– 2 – 2) – 1(2 + 6) – 6(1 – 3)

|A| = – 4 – 8 + 12

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = 6k

x – y = – 2k

**Question 5. Solve the system of homogeneous linear equations by matrix method:**

**x + y + z = 0**

**x – y – 5z = 0**

**x + 2y + 4z = 0**

**Solution:**

Given:

x + y + z = 0

x – y – 5z = 0

x + 2y + 4z = 0

A X = 0

Now, |A| = 1(6) – 1(9) + 1(3)

|A| = 9 – 9

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = –k

x – y = 5k

**Question 6. Solve the system of homogeneous linear equations by matrix method:**

**x + y – z = 0**

**x – 2y + z = 0**

**3x + 6y –5z = 0**

**Solution:**

Given:

x + y – z = 0

x – 2y + z = 0

3x + 6y –5z = 0

A X = 0

Now, |A| = 1(4) – 1(–8) – 1(12)

|A| = 4 + 8 – 12

|A| = 0

Hence, the system has infinite solutions

Let z = k

x + y = –k

x – 2y = –k

**Question 7. Solve the system of homogeneous linear equations by matrix method:**

**3x + y – 2z = 0**

**x + y + z = 0**

**x – 2y + z =0**

**Solution:**

Given:

3x + y – 2z = 0

x + y + z = 0

x – 2y + z =0

A X = 0

Now, |A| = 3(3) – 1(0) – 2(–3)

|A| = 9 – 0 + 6

|A| = 15 ≠ 0,

Hence, the given system has only trivial solutions given by x = y = z = 0.

**Question 8. Solve the system of homogeneous linear equations by matrix method:**

**2x + 3y – z =0**

**x – y – 2z = 0**

**3x + y + 3z = 0**

**Solution:**

Given:

2x + 3y – z =0

x – y – 2z = 0

3x + y + 3z = 0

A X = 0

Now, |A| = 2(–3 + 2) – 3(3 + 6) – 1(4)

|A| = –2 – 27 – 4

|A| = –33 ≠ 0,

Hence, the given system has only trivial solutions given by x = y = z = 0.