Evaluate the determinant of the matrix below: [[0, 4, 7], [1, 5, 8], [3, 6, 9]]

The matrix is:

$$A = \begin{pmatrix} 0 & 4 & 7 \\ 1 & 5 & 8 \\ 3 & 6 & 9 \end{pmatrix}$$

We will expand along the first row. The formula for the determinant of a $3 \times 3$ matrix $\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ is $a(ei - fh) - b(di - fg) + c(dh - eg)$.

$$\det(A) = 0 \cdot \det \begin{pmatrix} 5 & 8 \\ 6 & 9 \end{pmatrix} - 4 \cdot \det \begin{pmatrix} 1 & 8 \\ 3 & 9 \end{pmatrix} + 7 \cdot \det \begin{pmatrix} 1 & 5 \\ 3 & 6 \end{pmatrix}$$

For the first term:

$$0 \cdot ((5 \times 9) - (8 \times 6)) = 0 \cdot (45 - 48) = 0 \cdot (-3) = 0$$

For the second term:

$$-4 \cdot ((1 \times 9) - (8 \times 3)) = -4 \cdot (9 - 24) = -4 \cdot (-15) = 60$$

For the third term:

$$7 \cdot ((1 \times 6) - (5 \times 3)) = 7 \cdot (6 - 15) = 7 \cdot (-9) = -63$$

$$\det(A) = 0 + 60 - 63$$ $$\det(A) = -3$$

The determinant of the matrix is:

$$-3$$