Add the downward diagonal products and subtract the upward products

Add the downward diagonal products and subtract the upward products


H
Asked by 12 months ago
240 points

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15–18.Warning: This trick does not generalize in any reasonable way to $4\times4$ or larger matrices.

$$ \begin{vmatrix} 0 & 3 & 1 \\ 4 & -5 & 0 \\ 3 & 4 & 1 \\ \end{vmatrix} $$

Add the
hansel

1 Answer

Answered by 12 months ago
8.7k points

rewrite the matrix in the following way by copying the first two columns and placing them on the right side of the matrix,

$$ \begin{vmatrix} 0 & 3 & 1 \\ 4 & -5 & 0 \\ 3 & 4 & 1 \end{vmatrix} \begin{matrix} &0 & 3 \\ &4 & -5 \\ &3 & 4 \end{matrix} $$

now add the product of the downward diagonals minus the product of the upward diagonals in the following way,

$$(0)(-5)(1)+(3)(0)(3)+(1)(4)(4)-(3)(-5)(1)-(4)(0)(0)-(1)(4)(3)$$

$$=16+15-12=19$$

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Asked: 12 months ago

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