Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.

Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.

Question
Matrices
asked 2021-03-02
Zero Divisors If a and b are real or complex numbers such thal ab = O. then either a = 0 or b = 0. Does this property hold for matrices? That is, if A and Bare n x n matrices such that AB = 0. is il true lhat we must have A = 0 or B = 0? Prove lhe resull or find a counterexample.

Answers (1)

2021-03-03
Given: "If a and b are real or complex numbers such thal ab = 0. then either a = 0 or b = 0"
No, the above property does not hold for matrices.
That is, if A and B are n x n matrices such that AB = 0 then it need not imply that either A =0 or B = 0.
Step 2
Counter example:
Take A=\left(\begin{array}{c}0&1\\ 0&0\end{array}\right) \text{ and } B=\left(\begin{array}{c}0&0\\ 1&0\end{array}\right) clearly , \(A\neq 0 \text{ and } B\neq 0\)
\(\text{But } AB=0\)
0

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