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Learn Exercise 4.2 with Free Lessons & Tips

By using properties of determinants, show that:

(i) 

(ii) 

(i) 

Applying R1 → R1 − Rand R2 → R2 − R3, we have:

Applying R1 → R1 + R2, we have:

Expanding along C1, we have:

Hence, the given result is proved.

(ii) Let.

Applying C1 → C1 − Cand C2 → C2 − C3, we have:

Applying C1 → C1 + C2, we have:

Expanding along C1, we have:

Hence, the given result is proved.

Comments

Choose the correct answer.

Let A be a square matrix of order 3 × 3, then is equal to

A. B. C. D.  

Answer: C

A is a square matrix of order 3 × 3.

Hence, the correct answer is C.

Comments

By using properties of determinants, show that:

We have,

Here, the two rows R1 and R3 are identical.

∴Δ = 0.

Comments

Using the property of determinants and without expanding, prove that:

 

Comments

Using the property of determinants and without expanding, prove that:

Here, the two rows R1 and R3 are identical.

Δ = 0.

Comments

Using the property of determinants and without expanding, prove that:

 

Comments

Using the property of determinants and without expanding, prove that:

By applying C3 → C3 + C2, we have:

Here, two columns C1 and C3 are proportional.

Δ = 0.

Comments

Using the property of determinants and without expanding, prove that:

Applying R2 → R2 − R3, we have:

Applying R1 ↔R3 and R2 ↔R3, we have:

Applying R1 → R1 − R3, we have:

Applying R1 ↔R2 and R2 ↔R3, we have:

From (1), (2), and (3), we have:

Hence, the given result is proved.

Comments

By using properties of determinants, show that:

Applying R2 → R2 + R1 and R3 → R3 + R1, we have:

 

Comments

By using properties of determinants, show that:

(i)

(ii)

(i) Step 1 

R1 = R1 + R2 + R3

Step 2

(5x+4) common in R1, bring that out, you are left with all 1s in R1

Applying C2 → C2 − C1, C3 → C3 − C1, we have:

Step 3

R2 = R2 - 2x(R1)

R3 = R3 - 2x(R1)

Now matrix is :

(5x+4)[(1,1,1),(0,4-x,0),(0,0,4-x)]

=(5x+4)(4-x)(4-x)

Expanding along C3, we have:

Hence, the given result is proved.

 

 

(ii) Here you need to subtract y(R1) from R2 and R3.

Applying R1 → R1 + R+ R3, we have:

Applying C2 → C2 − Cand C3 → C3 − C1, we have:

Expanding along C3, we have:

Comments

By using properties of determinants, show that:

(i)

(ii)

(i)

Applying R1 → R1 + R2 + R3, we have:

Applying C2 → C2 − C1, C3 → C3 − C1, we have:

Expanding along C3, we have:

Hence, the given result is proved.

(ii)

Applying C1 → C1 + C2 + C3, we have:

Applying R2 → R2 − R1 and R3 → R3 − R1, we have:

Expanding along R3, we have:

Hence, the given result is proved.

Comments

By using properties of determinants, show that:

Applying R1 → R1 + R2 + R3, we have:

Applying C2 → C2 − C1 and C3 → C3 − C1, we have:

Expanding along R1, we have:

Hence, the given result is proved.

Comments

By using properties of determinants, show that:

Applying R1 → R1 + bR3 and R2 → R2aR3, we have:

Expanding along R1, we have:

 

Comments

By using properties of determinants, show that:

Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have:

Applying R2 → R2 − R1 and R3 → R3 − R1, we have:

Applying C1aC1, C2 bC2, and C3cC3, we have:

Expanding along R3, we have:

Hence, the given result is proved.

Comments

Which of the following is correct?

A. Determinant is a square matrix.

B. Determinant is a number associated to a matrix.

C. Determinant is a number associated to a square matrix.

D. None of these 

Answer: C

We know that to every square matrix, of order n. We can associate a number called the determinant of square matrix A, where element of A.

Thus, the determinant is a number associated to a square matrix.

Hence, the correct answer is C.

Comments

By using properties of determinants, show that:

Applying R2 → R2 − Rand R3 → R3 − R1, we have:

Applying R3 → R3 + R2, we have:

Expanding along R3, we have:

Hence, the given result is proved.

Comments

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