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in Determinants by (15.2k points)
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Evaluate :

\(\begin{bmatrix} 1^2& 2^2 & 3^2 \\[0.3em] 2^2 & 2^2 & 4^2 \\[0.3em] 3^2 &4^2 & 5^2 \end{bmatrix}\)

[(1^2,2^2,3^2)(2^2,2^2,4^2)(3^2,4^2,5^2)]

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\(\begin{bmatrix} 1^2& 2^2 & 3^2 \\[0.3em] 2^2 & 2^2 & 4^2 \\[0.3em] 3^2 &4^2 & 5^2 \end{bmatrix}\) = \(\begin{bmatrix} 1 &4 & 9 \\[0.3em] 4 & 9 & 16 \\[0.3em] 9 & 16 & 25 \end{bmatrix}\)

Expanding by first row, we get, 

1(9 × 25 - 16 × 16) + 4(16 × 9 - 4 × 25) + 9(4 × 16 - 9 × 9) = - 31 + 176 - 153 =- 8

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