Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column

# Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column

H
240 points

Compute the determinants in Exercises 1–8 using a cofactor expansion across the first row. In Exercises 1–4, also compute the determinant by a cofactor expansion down the second column.

$$\begin{vmatrix} 1 & 2 & 4 \\ 3 & 1 & 1 \\ 2 & 4 & 2 \\ \end{vmatrix}$$

Compute the
hansel

8.7k points

across the first row

$$= (-1)^{1+1} (1) \begin{vmatrix} 1 & 1 \\ 4 & 2 \\ \end{vmatrix} + (-1)^{1+2} (2) \begin{vmatrix} 3 & 1 \\ 2 & 2 \\ \end{vmatrix} + (-1)^{1+3} (4) \begin{vmatrix} 3 & 1 \\ 2 & 4 \\ \end{vmatrix}$$

$$= (1)(2-4)+(-2)(6-2)+(4)(12-2)=-2-8+40=30$$

down the second column

$$= (-1)^{1+2} (2) \begin{vmatrix} 3 & 1 \\ 2 & 2 \\ \end{vmatrix} + (-1)^{2+2} (1) \begin{vmatrix} 1 & 4 \\ 2 & 2 \\ \end{vmatrix} + (-1)^{3+2} (4) \begin{vmatrix} 1 & 4 \\ 3 & 1 \\ \end{vmatrix}$$

$$= (-2)(6-2)+(1)(2-8)+(-4)(1-12)=-8-6+44=30$$

R
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