Count of 3×3 Matrices with Trace of AAᵀ = 3 | Matrices | JEE Maths PYQ | Doubtify JEE

 

🔢 Matrices & Determinants | JEE Maths | Doubtify JEE

---

📌 Question:

The number of all 3 × 3 matrices A, with entries from the set {-1, 0, 1}, such that the sum of the diagonal elements of AAᵀ is 3, is:

---

🖼️ Question Image:

🧠 Solution Image:

---

📽️ Video Solution:

---

🧠 Concept Behind the Problem:

This question is based on two core concepts in matrices:

1. Matrix multiplication and transpose: AAᵀ is a symmetric matrix, and its diagonal elements represent the sum of squares of each row vector of matrix A.

2. Trace of a matrix: The sum of diagonal elements of AAᵀ is equal to the trace, which in this case must be 3.

---

🔍 Why this Question is Important:

• ✅ Tests understanding of matrix operations and properties of transpose.

• ✅ Reinforces the idea of trace and how it relates to inner product norms of rows.

• ✅ A good blend of combinatorics and algebraic logic — perfect JEE style.

• ✅ Frequently asked format in JEE Advanced and Olympiad-level practice.

---

🧠 Pro Tip:

Always remember:
For any matrix A, Tr(AAᵀ) = sum of squares of all entries in A’s rows (i.e., ||Row1||² + ||Row2||² + ||Row3||²)

So, Tr(AAᵀ) = total number of non-zero elements (±1) squared.
Thus, this question is reduced to counting 3×3 matrices with exactly 3 entries as ±1 and the rest 0.

---

📩 Have a Doubt?

DM us on Instagram or mail at doubtifyqueries@gmail.com.
We reply within 24 hours. 💬

---