GATE CS 2020 Q.37 (CS Q.27): Linear Algebra

Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let $\text{rank}(M)$ and $\det(M)$ denote the rank and determinant of a matrix $M$ , respectively. Consider the following statements.

I. $\text{rank}(AB) = \text{rank}(A) \cdot \text{rank}(B)$
II. $\det(AB) = \det(A) \cdot \det(B)$
III. $\text{rank}(A + B) \le \text{rank}(A) + \text{rank}(B)$
IV. $\det(A + B) \le \det(A) + \det(B)$

Which of the above statements are TRUE?

I and II only

I and IV only

II and III only

III and IV only

Official answer and solution

Official answer: C
From: Official GATE 2020 answer key

Solution

Evaluate standard linear algebra properties for matrix rank and determinant.

Statement I: $\text{rank}(AB) = \text{rank}(A) \cdot \text{rank}(B)$

Counterexample: Let $A = I_2$ and $B = I_2$ (identity matrices of size $2 \times 2$ ).
$\text{rank}(A) = 2$ , $\text{rank}(B) = 2$ , so $\text{rank}(A) \cdot \text{rank}(B) = 4$ .
But $AB = I_2$ which has rank 2, not 4. Statement I is FALSE.

Statement II: $\det(AB) = \det(A) \cdot \det(B)$

This is a fundamental theorem of determinants for square matrices over any field. Statement II is TRUE.

Statement III: $\text{rank}(A + B) \le \text{rank}(A) + \text{rank}(B)$

The column space of $A + B$ is contained within the sum of the column spaces of $A$ and $B$ : $\text{col}(A+B) \subseteq \text{col}(A) + \text{col}(B)$ .
By dimension of subspaces: $\dim(\text{col}(A+B)) \le \dim(\text{col}(A)) + \dim(\text{col}(B))$ . Statement III is TRUE.

Statement IV: $\det(A + B) \le \det(A) + \det(B)$

Counterexample: Let $A = I_2$ and $B = I_2$ .
$\det(A) = 1$ , $\det(B) = 1$ , so $\det(A) + \det(B) = 2$ .
$A + B = 2I_2$ , so $\det(A + B) = \det(2I_2) = 2^2 \cdot 1 = 4$ .
Since $4 \not\le 2$ , Statement IV is FALSE.

Therefore, statements II and III only are TRUE.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2020 Computer Science and Information Technology (CS)
Question: 37 of 65, printed as CS Q.27
Section: Computer Science and Information Technology

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