matrices

Question

If A is a 3×3 matrix and |A| = 5, then |3A| equals

RankGuru · Accepted Answer

1. **Scalar Property**: For an $n 	imes n$ matrix $A$ and a scalar $k$, the property of determinants is $|kA| = k^n |A|$.
2. **Identify Values**: Given order $n = 3$, scalar $k = 3$, and determinant $|A| = 5$.
3. **Calculate**:
   - $|3A| = 3^3 	imes |A|$
   - $|3A| = 27 	imes 5$
4. **Result**: $135$.
- Reason: Multiplying a matrix by a constant multiplies every row, and since there are 3 rows, the constant factor comes out as $k \cdot k \cdot k$.

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