Matrix Entries Problem with Diagonal Sum Constraint

❓ Question

The number of all $3\times3$ matrices $A$, with entries from the set

$$\{-1,0,1\}$$

such that the sum of the diagonal elements of

$$AA^T$$

is $3$, is ________.

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🖼 Question Image

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✍️ Short Explanation

This problem is based on:

👉 Properties of

$$AA^T$$

👉 Trace of matrix
👉 Counting arrangements.

Main idea:

Diagonal elements of

$$AA^T$$

represent sum of squares of entries of rows.

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🔷 Step 1 — Understand Diagonal of $AA^T$ 💯

For any matrix:

$$AA^T$$

Diagonal entries are:

$$a_{i1}^2+a_{i2}^2+a_{i3}^2$$

So sum of diagonal elements:

$$\text{Trace}(AA^T)$$

equals sum of squares of all entries of matrix $A$.

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🔷 Step 2 — Use Given Condition

Given:

$$\text{Trace}(AA^T)=3$$

Entries allowed are:

$$-1,0,1$$

Their squares are:

$$1,0,1$$

So total sum of squares being $3$ means:

👉 Exactly three entries are non-zero
👉 Remaining six entries are zero.

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🔷 Step 3 — Count Positions

Total entries in matrix:

$$3\times3=9$$

Choose positions of three non-zero entries:

$${}^9C_3$$
$$=84$$

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🔷 Step 4 — Fill Values

Each chosen position can contain:

$$1 \text{ or } -1$$

So for 3 positions:

$$2^3$$
$$=8$$

ways.

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🔷 Step 5 — Total Matrices

$$84\times8$$
$$=672$$

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🔷 Step 6 — JEE Trap Alert 🚨

❌ Trace ko determinant samajh lena

❌ $AA^T$ multiply karne lag jaana fully

❌ Zero entries counting miss kar dena

Remember:

$$\text{Trace}(AA^T) = \text{sum of squares of all entries}$$

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✅ Final Answer

$$\boxed{672}$$

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📚 Related Topics

• Matrix Entries Problem with Diagonal Sum Constraint
• Determinant Zero Method for Matrix Counting
• Adjugate of Adjugate Matrix Determinant Problem