rank(AB) = rank(A) if B is invertible

For any two matrices such that $AB$ makes sense, $$\DeclareMathOperator{\rk}{rk} \rk(AB)\le\rk(A) $$ If $B$ is invertible, then $$ \rk(A)=\rk(ABB^{-1})\le\rk(AB)\le\rk(A) $$


The rank is the dimension of the column space. The column space of $AB$ is the same as the column space of $A$.

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Linear Algebra

Matrices

Matrix Rank

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