adj(A) & Determinant Power Rules in 59 Seconds! 🔥 | JEE Maths

 

❓ Concept

🎬 Adj(A) & Determinant Tricks

Adjugate (adj) questions look scary in JEE — but in reality, sab kuch power rules pe based hota hai.
Agar 4 identities yaad hain, toh 10 seconds mein answer.

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1️⃣ What is adj(A)?

Adjugate of a matrix $A$:

$$\text{adj}(A) = \text{transpose of cofactor matrix}$$

⭐ Most Important Identity:

$$A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = |A|\,I$$

📌 Ye identity almost har JEE matrix question ka base hoti hai.

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2️⃣ adj(A) for Invertible A

If:

$$|A| \ne 0$$

Then:

$$\text{adj}(A) = |A|\,A^{-1}$$

👉 Matlab adj(A) behaves like det(A) × inverse.

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3️⃣ Determinant of adj(A) — BIG TRICK 🔥

For any $n \times n$ matrix:

$$|\text{adj}(A)| = |A|^{\,n-1}$$

📌 Very powerful rule:

• adj lene se determinant power mein convert ho jaata hai

For JEE (most common):

• $n = 3$
  

$$|\text{adj}(A)| = |A|^2$$

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4️⃣ adj(adj(A)) & Power Rules

General identity (for $n > 1$):

$$\text{adj}(\text{adj}(A)) = |A|^{\,n-2}\,A$$

Now take determinant on both sides:

$$|\text{adj}(\text{adj}(A))| = |A|^{(n-1)^2}$$

📌 Yahin se tricky exponent patterns aate hain.

For $n = 3$:

$$|\text{adj}(\text{adj}(A))| = |A|^4$$

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5️⃣ JEE Tips — How to Apply FAST ⚡

Follow this fixed checklist 👇

✅ Step 1

Check:

$$|A| = 0 \quad \text{or} \quad |A| \ne 0$$

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✅ Step 2

Use:

$$A\cdot \text{adj}(A) = |A|I$$

to shift between $A$ and $\text{adj}(A)$

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✅ Step 3

If powers involved:
👉 Take determinant on both sides

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✅ Step 4 (Most Important)

For 3×3 matrices, directly remember:

$$|\text{adj}(A)| = |A|^2$$
$$|\text{adj}(\text{adj}(A))| = |A|^4$$

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✅ Final Takeaway

🧠 Adj + Det Master Rule:

• adj = det power game

• det(adj(A)) = det(A)ⁿ⁻¹

• adj(adj(A)) gives higher powers

• No need to expand matrices ever

Once these are clear, matrix questions become free marks.