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# 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

Linear Algebra
Medium
##### Suggested Book
Linear algebra, Friedberg, Insel

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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