Question

Let \(A\) and \(B\) be \(n\times n\) real matrices. Which of the following statements is correct?

• A. \(rank(A+B)=rank(A)+rank(B)\)
• B. \(rank(A+B)\le rank(A)+rank(B)\)
• C. \(rank(A+B)=\min\{rank(A),rank(B)\}\)
• D. \(rank(A+B)=\max\{rank(A),rank(B)\}\)

Hint:

Remember the fundamental rank inequality: \[ rank(A+B)\le rank(A)+rank(B) \] and \[ rank(AB)\le \min\{rank(A),rank(B)\}. \] These two formulas are among the most frequently used results in matrix theory.

Answer 1

The Correct Option is B

Concept: Rank of a matrix is the maximum number of linearly independent rows or columns present in the matrix. One of the most important inequalities involving ranks of matrices is the rank inequality: \[ rank(A+B)\le rank(A)+rank(B) \] This inequality is valid for matrices of the same order and is extensively used in Linear Algebra. The equality sign does not necessarily hold for all matrices. Therefore only the inequality form is universally true.

Step 1: Recall the rank inequality theorem.
For any two matrices \(A\) and \(B\) of the same order, \[ rank(A+B)\le rank(A)+rank(B) \] This result follows from the fact that the column space of \(A+B\) is contained in the sum of the column spaces of \(A\) and \(B\).

Step 2: Check option (A).
The statement \[ rank(A+B)=rank(A)+rank(B) \] is not always true. For example, if \(A=B\), then \[ rank(A+B)=rank(2A)=rank(A), \] which generally is not equal to \[ rank(A)+rank(B)=2\,rank(A). \] Hence option (A) is false.

Step 3: Check options (C) and (D).
The rank of \(A+B\) may be smaller or larger than the minimum rank and need not be equal to the maximum rank. Therefore neither of these relations is universally valid.

Step 4: Identify the correct statement.
The only statement that is always true for all matrices \(A\) and \(B\) is \[ rank(A+B)\le rank(A)+rank(B). \] \[ \boxed{rank(A+B)\le rank(A)+rank(B)} \] Hence option (B) is correct.