The GATE Grind

GATE 2020 CS – Question 37

Engineering Mathematics · Linear Algebra · 2 marks · Multiple choice

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. $\text{rank}(AB)=\text{rank}(A)\,\text{rank}(B)$

II. $\det(AB)=\det(A)\det(B)$

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?

  1. I and II only
  2. I and IV only
  3. II and III only
  4. III and IV only

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Correct answer: (C) II and III only

Explanation

II (multiplicativity of determinant) and III (subadditivity of rank) are standard theorems. I is false, e.g. $A=B=\text{diag}(1,0)$ gives rank(AB)=1 but the product of ranks is 1, and $A=B=\begin{pmatrix}0&1\\0&0\end{pmatrix}$ gives rank(AB)=0 versus 1. IV is false, e.g. $A=B=I_2$ gives 4 > 2.