The determinant $$\begin{vmatrix}\mathbf{1} & \mathbf{3} & \mathbf{7} \\ \mathbf{4} & \mathbf{9} & \mathbf{1} \\ \mathbf{2} & \mathbf{7} & \mathbf{6}\end{vmatrix}$$ is

$$1 \times \begin{vmatrix} 9 & 1 \\ 7 & 6 \end{vmatrix} = 1 \times (9 \times 6 - 7 \times 1) = 1 \times (54 - 7) = \mathbf{47}$$
$$-3 \times \begin{vmatrix} 4 & 1 \\ 2 & 6 \end{vmatrix} = -3 \times (4 \times 6 - 2 \times 1) = -3 \times (24 - 2) = -3 \times 22 = \mathbf{-66}$$
$$7 \times \begin{vmatrix} 4 & 9 \\ 2 & 7 \end{vmatrix} = 7 \times (4 \times 7 - 2 \times 9) = 7 \times (28 - 18) = 7 \times 10 = \mathbf{70}$$
The determinant = 47 - 66 + 70 = 51.