Test: Determinants - 1 - JEE MCQ

Apply C₂ → C₂ + C_3,

Apply , R₁ → R₁+R₂+R₃,

Apply , C₃→ C₃ - C₁, C₂^→C₂ - C₁,

=(a+b+c)³

Commutative law, in mathematics, either of two laws relating to number operations of addition and multiplication, stated symbolically: a + b = b + a and ab = ba

The determinant of a matrix A and its transpose always same.

, Apply , C₂ → C₂ + C₃





= 0, (∵ C₁ = C₂)

⇒ (1-x)(2-x)(3-x) = 0 ⇒x = 1,2,3

Let A be a square matrix of order 2 . then ,|adj.A| = |A|

If A is a symmetric matrix then by definition AT=A
Option A is correct.

, because , row 1 and row 3 are identical.

Apply , C1→C1 - C3, C₂→C₂-C₃

= 10 - 12 = -2

If A is anon singular matrix of order , then 

Det.(A+B) ≠ Det.A + Det.B.

Correct option is D.

If A B be two square matrices such that AB = O, then, 

Apply , C₁ → C₁ - C₂, C₂ → C₂ - C₃,

Because here row 1 and 2 are identical

Because , the determinant of a skew symmetric matrix of odd order is always zero and of even order is a non zero perfect square.

Because , the inverse of an identity matrix is an identity matrix.

By the property of transpose of a matrix ,(AB)’ = B’A’.

Only non-singular matrices possess inverse.

Apply, C₁→ C₁+ C₂+C₃+C₄,

Apply, R1 →R₁ - R₂,

Apply, R₁→ R₁ - R₂, R₂ → R₂ - R₃

=(x+3a) (a -x)³ (1) = (x+3a)(a-x)³

Greatest value = 2 

Operate,

Apply R₃→R₃- R₁, R₂→ R₂ -R₁,

⇒ -6(5x² - 20) +15(2x-4) = 0

⇒ (x- 2)(x+1) = 0⇒x=2, -1

⇒-6(5x² - 20) + 15(2x-4) = 0

⇒(x-2)(x+1) = 0 ⇒ x=2, -1

N = 2 ×3 × 4 = 24.