Let L be lower triangular, U be upper triangular, and D be diagonal.
a. If L and U are both unit triangular and LDU is diagonal, does it follow that L and U are diagonal?
b. If LDU is nonsingular and diagonal, does it follow that L and U are diagonal?
c. If L and U are both unit triangular and if LDU is diagonal, does it follow that L = U = I?
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