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If-a-matrix-A-is-such-that-3A-3-2A-2-5A-I-0-then-A-1-is-equal-to-




Question Number 70383 by ®Ëƒ ¬Ë°¹¾¨ ‰¦Í¦¿¨ ¸Ë¹Ç² last updated on 04/Oct/19
If a matrix A is such that 3A^3 +2A^2 +5A+I=0,  then A^(−1) is equal to
$$\mathrm{If}\:\mathrm{a}\:\mathrm{matrix}\:{A}\:\mathrm{is}\:\mathrm{such}\:\mathrm{that}\:\mathrm{3}{A}^{\mathrm{3}} +\mathrm{2}{A}^{\mathrm{2}} +\mathrm{5}{A}+{I}=\mathrm{0}, \\ $$$$\mathrm{then}\:{A}^{−\mathrm{1}} \mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$
Commented by mathmax by abdo last updated on 04/Oct/19
3A^3  +2A^2  +5A +I =0 ⇒3A^3  +2A^2  +5A=−I ⇒  A.(−3A^2 −2A −5I) =I ⇒ A^(−1) =−3A^2 −2A−5I .
$$\mathrm{3}{A}^{\mathrm{3}} \:+\mathrm{2}{A}^{\mathrm{2}} \:+\mathrm{5}{A}\:+{I}\:=\mathrm{0}\:\Rightarrow\mathrm{3}{A}^{\mathrm{3}} \:+\mathrm{2}{A}^{\mathrm{2}} \:+\mathrm{5}{A}=−{I}\:\Rightarrow \\ $$$${A}.\left(−\mathrm{3}{A}^{\mathrm{2}} −\mathrm{2}{A}\:−\mathrm{5}{I}\right)\:={I}\:\Rightarrow\:{A}^{−\mathrm{1}} =−\mathrm{3}{A}^{\mathrm{2}} −\mathrm{2}{A}−\mathrm{5}{I}\:. \\ $$

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