Skill: Expand a determinant along a specified row

To expand a $3\times3$ determinant along a row, use each entry in that row with its matching $2\times2$ minor. For the first row, the cofactor signs are $+,-,+$. To form a minor, delete the chosen entry’s row and column. Then evaluate each $2\times2$ determinant using $ad-bc$.

Expand this determinant along the first row:

$$\begin{vmatrix} 2 & 1 & 3\\ 0 & -1 & 4\\ 5 & 2 & 1 \end{vmatrix}$$

The first-row entries are $2$, $1$, and $3$.

• For $2$, delete the first row and first column:

$$\begin{vmatrix}-1&4\\2&1\end{vmatrix} =(-1)(1)-(4)(2)=-9$$

• For $1$, delete the first row and second column:

$$\begin{vmatrix}0&4\\5&1\end{vmatrix} =(0)(1)-(4)(5)=-20$$

• For $3$, delete the first row and third column:

$$\begin{vmatrix}0&-1\\5&2\end{vmatrix} =(0)(2)-(-1)(5)=5$$

Apply the $+,-,+$ signs and multiply by the corresponding first-row entries:

$$2(-9)-1(-20)+3(5)=-18+20+15=\boxed{17}$$

The determinant is $17$.