Add the downward diagonal products and subtract the upward products

# Add the downward diagonal products and subtract the upward products

H
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.

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

hansel

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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