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Let-A-and-B-are-3-3-matrices-The-statements-below-which-is-True-is-A-AB-BA-B-If-AB-0-then-only-A-0-or-B-0-is-true-C-If-AB-I-then-only-A-I-or-A-I-is-true-D-There-exist-A-

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Question Number 35482 by Joel579 last updated on 19/May/18
Let A and B are 3 ×3 matrices. The statements below which is True is ... (A) AB = BA (B) If AB = 0, then only A = 0 or B = 0 is true (C) If AB = I, then only A = I or A = −I is true (D) There exist A where A ≠ 0, and yet A^2 = 0 (E) ∣A + B∣ = ∣A∣ + ∣B∣
$$\mathrm{Let}\:{A}\:\mathrm{and}\:{B}\:\mathrm{are}\:\mathrm{3}\:×\mathrm{3}\:\mathrm{matrices}.\:\mathrm{The}\:\mathrm{statements}\:\mathrm{below} \\ $$$$\mathrm{which}\:\mathrm{is}\:{True}\:\mathrm{is}\:… \\ $$$$\left(\mathrm{A}\right)\:{AB}\:=\:{BA} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{If}\:{AB}\:=\:\mathrm{0},\:\mathrm{then}\:\mathrm{only}\:{A}\:=\:\mathrm{0}\:\mathrm{or}\:{B}\:=\:\mathrm{0}\:\mathrm{is}\:\mathrm{true} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{If}\:{AB}\:=\:{I},\:\mathrm{then}\:\mathrm{only}\:{A}\:=\:{I}\:\mathrm{or}\:{A}\:=\:−{I}\:\mathrm{is}\:\mathrm{true} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{There}\:\mathrm{exist}\:{A}\:\mathrm{where}\:{A}\:\neq\:\mathrm{0},\:\mathrm{and}\:\mathrm{yet}\:{A}^{\mathrm{2}} \:=\:\mathrm{0} \\ $$$$\left(\mathrm{E}\right)\:\mid{A}\:+\:{B}\mid\:=\:\mid{A}\mid\:+\:\mid{B}\mid \\ $$
Commented by candre last updated on 19/May/18
A is false because miltiplication don′t comute in gemeral=
$$\mathrm{A}\:\mathrm{is}\:\mathrm{false}\:\mathrm{because}\:\mathrm{miltiplication}\:\mathrm{don}'\mathrm{t}\:\mathrm{comute}\:\mathrm{in}\:\mathrm{gemeral}= \\ $$

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