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if the equation $AB +5I_n = 3A + 2B$ holds then which is affirmation is true :

a ) $ A -2I_n$ is invertible

b)$B-3I_n$ is invertible

PS : I don't understand where to even begin with this problem

Serbacul
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    "I have no clue" questions are not well received here. – markvs Feb 26 '22 at 11:01
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    Invertible means the determinant is non-zero. Use this fact. Note that (a-2)(b-3) = ab - 2b - 3a + 5. Commutativity of matrix multiplication is important – egglog Feb 26 '22 at 11:01
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    but since i dont know how to start this problem i kinda need someone to explain it dont i ? – Serbacul Feb 26 '22 at 11:06
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    Strat by writing down the definition of invertibility for $A-2I_n$. Look up the determinant. One can always try something. Start doing it! Also the title is not good. You have tried more for your recent question. – Dietrich Burde Feb 26 '22 at 11:57
  • If you don't understand the question, go back and ask about the definitions and concepts required to do so. Just throwing up a question and saying "I have no idea" makes it pointless to both the community and to you. – Nij Feb 27 '22 at 00:00

1 Answers1

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As elaborated in the comment, consider $$(A-2I)(B-3I)=AB-3A-2B+6I=(?)$$ Then what can you say about $A-2I$ and $B-3I$? (There is nothing to do with the determinants here!)

One more interesting question, can you use this to show that $AB=BA$?

Bernard Pan
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