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Determinants

ICSE · Class 12 · Mathematics

Flashcards for Determinants — ICSE Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.

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A visual representation defining a determinant as a scalar value associated with a square matrix, showing its notation and basic structure.
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30 Flashcards
Card 1Determinant of order 2

Evaluate the determinant: |A| for A = [[x, x+1], [x-1, x]].

Answer

Step 1: Use the 2x2 formula |A| = a11a22 - a21a12. Step 2: |A| = x·x - (x-1)(x+1). Step 3: |A| = x^2 - (x^2 - 1). Step 4: |A| = 1. Answer: 1…

Card 2Trigonometric determinant

Evaluate: [[cos 15°, sin 15°], [sin 75°, cos 75°]].

Answer

Step 1: Use |A| = a11a22 - a21a12. Step 2: |A| = cos 15°·cos 75° - sin 15°·sin 75°. Step 3: Use cos(A+B) = cos A cos B - sin A sin B. Step 4: |A| = cos(15° + 75°) = cos 90°. Step 5: cos 90° = 0. Answe…

Card 3Determinant of order 3

Evaluate the determinant: [[2, -1, -2], [0, 2, -1], [3, -5, 0]].

Answer

Step 1: Expand along the first row. Step 2: |A| = 2[(2)(0) - (-5)(-1)] - (-1)[(0)(0) - (3)(-1)] + (-2)[(0)(-5) - (3)(2)]. Step 3: |A| = 2(0 - 5) + 1(0 + 3) - 2(0 - 6). Step 4: |A| = -10 + 3 + 12. Step…

Card 4Equation using determinants

Find the value of x if [[2, 4], [5, 1]] = [[2x, 4], [6, x]].

Answer

Step 1: Find the determinant of both sides. Step 2: Left side = 2·1 - 5·4 = 2 - 20 = -18. Step 3: Right side = (2x)(x) - 6·4 = 2x^2 - 24. Step 4: Set equal: 2x^2 - 24 = -18. Step 5: 2x^2 = 6. Step 6: …

Card 5Minor

Find the minors of the elements of [[1, -2], [4, 3]].

Answer

Step 1: Delete the row and column of each element. Step 2: M11 = 3. Step 3: M12 = 4. Step 4: M21 = -2. Step 5: M22 = 1. Answer: M11 = 3, M12 = 4, M21 = -2, M22 = 1…

Card 6Cofactor

Find the cofactors of the elements of [[1, -2], [4, 3]].

Answer

Step 1: Use Cij = (-1)^(i+j) Mij. Step 2: C11 = (+1)(3) = 3. Step 3: C12 = (-1)(4) = -4. Step 4: C21 = (-1)(-2) = 2. Step 5: C22 = (+1)(1) = 1. Answer: C11 = 3, C12 = -4, C21 = 2, C22 = 1…

Card 7Laplace expansion

Why is the determinant equal to the sum of elements of any row multiplied by their cofactors?

Answer

This is Laplace expansion. For a row or column, |A| = sum of products of elements with corresponding cofactors. Example for a 2x2 matrix [[a, b], [c, d]]: |A| = aC11 + bC12 = a(d) + b(-c) = ad - bc. S…

Card 8Cofactor expansion

Using cofactors of the first column, evaluate [[4, 1], [-3, 2]].

Answer

Step 1: First column elements are 4 and -3. Step 2: C11 = 2. Step 3: C21 = -1. Step 4: |A| = 4·2 + (-3)(-1). Step 5: |A| = 8 + 3 = 11. Answer: 11…

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What are the important topics in Determinants for ICSE Class 12 Mathematics?
Key topics in Determinants include Determinants: Complete Concept Map, Complete Mind Map of Determinants Chapter, Determinants - Concept Hierarchy and Structure. These are the concepts ICSE Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Determinants — ICSE Class 12 Mathematics?
Understand the core concepts first, then work through the 100 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
How many flashcards are available for Determinants?
There are 30 flashcards for Determinants covering key definitions, formulas, and concepts. Use them daily for 10–15 minutes for best results.

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