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Linear Algebra

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Lower Triangular and not Maximal Rank

LINALG-1CS7EC

Suppose $A$ is a $6\times6$ lower triangular matrix of rank $5$. Which of the following statements is necessarily true?

A

$A$ has at least one zero on its main diagonal

B

$A$ is diagonal

C

$A$ is symmetric

D

The main diagonal entries of $A$ are either all positive or all negative

E

The null space of $A$ has dimension $0$