Q45 / 160 · TS EAMCET 2024 (09 May Shift 1) · Mathematics

If $A$ is square matrix and $A^2+I=2 A$, then $A^9=$

  • A
    $8 A^2-7 I$
  • B
    $9 A+8 I$
  • Correct answer
    C
    $9 A-8 I$
  • D
    $8 A^2+7 I$

Solution

$A^2=2 A-I$ $\begin{aligned} A^3 & =2 A^2-A=2(2 A-I)-A=3 A-2 I \\ A^6 & =A^3 \cdot A^3=(3 A-2 I)(3 A-2 I) \\ & =9 A^2-12 A+4 I \\ & =9(2 A-I)-12 A+4 I=6 A-5 I . \\ A^9 & =A^6 \cdot A^3=(6 A-5 I) \cdot(3 A-2 I) \\ & =18 A^2-27 A+10 I \\ & =18(2 A-I)-27 A+10 I=9 A-8 I .\end{aligned}$

Hence, the correct answer is option: 'C'.

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