Mathematics (Calculus and Linear Algebra)
Mathematics (Calculus and Linear Algebra)
This domain tests the calculus and linear-algebra tools used throughout actuarial work: differentiation, integration, series, matrix operations, and eigenvalues. Items are usually short but unforgiving of sign and boundary errors.
Core rules and concepts
- Derivative rules: product, quotient, chain; d/dx ln x=1/x, d/dx e^{ax}=a e^{ax}.
- Integration by parts ∫u dv=uv−∫v du; substitution reverses the chain rule.
- Taylor expansion about a: f(a)+f′(a)(x−a)+f″(a)(x−a)²/2!+…
- Matrix inverse exists iff det(A)≠0; (AB)^{-1}=B^{-1}A^{-1}.
- Eigenvalues solve det(A−λI)=0; eigenvectors solve (A−λI)v=0.
- A symmetric matrix is positive definite iff all leading principal minors are positive.
- Ax=b has a unique solution when A is invertible; otherwise check consistency.
Worked example
For A=[[2,1],[1,2]], det(A−λI)=(2−λ)²−1=λ²−4λ+3=(λ−1)(λ−3). Eigenvalues are 1 and 3. For λ=1, (A−I)v=[[1,1],[1,1]]v=0 gives v=(1,−1). For λ=3, v=(1,1). Both eigenvalues are positive, so A is positive definite.
Traps that cost marks
- Dropping the inner derivative in a chain-rule calculation.
- Forgetting the constant or mis-substituting limits in a definite integral.
- Assuming det(A+B)=det(A)+det(B).
- Treating a repeated eigenvalue as automatically diagonalizable.
- Reversing the order in (AB)^{-1}.