Prerequisite Qualification: Matrix :Introduction to Financial Engineering and Risk Management (Introduction to Financial Engineering and Risk Management) Answers 2025

Question 1

Correct statements:

✅ For identity matrix, we have $AI=IA=A$

❌ If $C=AA$, then CT=CC^T = CCT=C (not always true)

✅ For matrix transpose, (AB)T=BTAT(AB)^T = B^T A^T(AB)T=BTAT

✅ If C=ATAC = A^T AC=ATA, then CT=CC^T = CCT=C

❌ Matrix multiplication is interchangeable AB≠BAAB \neq BAAB=BA in general

Question 2

Correct statements:

❌ $g(x)=2x+c$ is not linear if c≠0c \neq 0c=0

✅ Given
v3=v1+v2v_3 = v_1 + v_2v3​=v1​+v2​
So
f(v3)=f(v1)+f(v2)=1+(−3)=−2f(v_3)=f(v_1)+f(v_2)=1+(-3)=-2f(v3​)=f(v1​)+f(v2​)=1+(−3)=−2
❌ Therefore statement saying f(v3)=−1f(v_3)=-1f(v3​)=−1 is false

✅ For zero vectors:
f(0n)=0mf(0_n)=0_mf(0n​)=0m​

✅ There exists a matrix A∈Rm×nA \in \mathbb{R}^{m\times n}A∈Rm×n such that
$f(x)=Ax$

Question 3

Matrix:

A=[111101212]A=\begin{bmatrix} 1 & 1 & 1\\ 1 & 0 & 1\\ 2 & 1 & 2 \end{bmatrix}A=​112​101​112​​

Row reduction shows only two independent rows.

✅ Rank of $A$ = 2

Question 4

Correct statements:

✅ Column rank of $A$ = Row rank of $A$

❌ If rank$(A)=m$, then range(A)=Rm(A)=\mathbb{R}^m(A)=Rm
(true only if mapping is onto; not always)

❌ Column rank ≠ row rank (false)

❌ If $m=n$, rank$(A)=n$ does imply $A$ is invertible
(statement says “cannot imply”, so false)

Final Summary Table