GATE CS 2020 Q27 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Let us analyze each statement individually to determine its validity:

• Statement I: $\text{rank}(AB) = \text{rank}(A) \text{rank}(B)$ (FALSE)
  A standard property of matrix rank is that $\text{rank}(AB) \le \min(\text{rank}(A), \text{rank}(B))$.
  Counterexample: Let $A = I_2$ and $B = I_2$ (the $2 \times 2$ identity matrix).
  $\text{rank}(A) = 2$ and $\text{rank}(B) = 2$.
  Then, $AB = I_2$, so $\text{rank}(AB) = 2$. However, $\text{rank}(A) \times \text{rank}(B) = 2 \times 2 = 4$. Since $2 \neq 4$, the statement is false.


• Statement II: $\text{det}(AB) = \text{det}(A) \text{det}(B)$ (TRUE)
  This is a fundamental multiplicative property of determinants for any two $n \times n$ matrices.


• Statement III: $\text{rank}(A + B) \le \text{rank}(A) + \text{rank}(B)$ (TRUE)
  This is the subadditivity property of matrix rank. Since the column space of $A + B$ is a subspace of the sum of the column spaces of $A$ and $B$, we have: $$\dim(\text{Col}(A+B)) \le \dim(\text{Col}(A) + \text{Col}(B)) \le \dim(\text{Col}(A)) + \dim(\text{Col}(B))$$ Thus, $\text{rank}(A+B) \le \text{rank}(A) + \text{rank}(B)$ always holds.


• Statement IV: $\text{det}(A + B) \le \text{det}(A) + \text{det}(B)$ (FALSE)
  There is no such general inequality for determinants.
  Counterexample: Let $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.
  $\text{det}(A) = 1$ and $\text{det}(B) = 1$, so $\text{det}(A) + \text{det}(B) = 2$.
  But $A + B = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$, which gives $\text{det}(A+B) = 4$.
  Since $4 \not\le 2$, the statement is false.

Therefore, only statements II and III are TRUE.