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ABC of rank: TIFR GS 2019, Part B Problem 5

Understand the problem

Suppose A, B, C are 3 \times 3 real matrices with Rank(A)=2, Rank(B)=1, Rank(C)=2.
Then Rank(ABC)=1.

We have to find out whether the statement is true or false.

Source of the problem

TIFR GS 2019, Part B Problem 5

Topic
Linear Algebra
Difficulty Level
Medium
Suggested Book
Linear algebra, Friedberg, Insel

Start with hints

Do you really need a hint? Try it first!

Can you try it with diagonal matrices with 1 or 2 non-zero diagonal entries?
Consider A with all zero entries except a_{11} and a_{22} (rank(A)=2). B is all zeros except b_{33} (rank(B)=1). AB= 0, thus ABC=0 and has rank 0.

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