Determinant:

• Denoted $\det(A)$ or $|A|$
• Measures the "volume scaling factor" of the linear transformation
• $\det(A) = 0$ if and only if $A$ is singular (non-invertible)
• For eigenvalues $\lambda_1, \ldots, \lambda_n$: $\det(A) = \lambda_1 \cdots \lambda_n$

Trace:

• Denoted $\text{tr}(A)$
• Sum of diagonal entries: $\text{tr}(A) = a_{11} + a_{22} + \cdots + a_{nn}$
• For eigenvalues $\lambda_1, \ldots, \lambda_n$: $\text{tr}(A) = \lambda_1 + \cdots + \lambda_n$