Let <math><mrow><mi>M</mi><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mo>−</mo><mi>α</mi></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mi>α</mi></mtd><mtd columnalign="center"><mn>0</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where is a non-zero real number an <math><mrow><mi>N</mi><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>4</mn><mn>9</mn></mrow></munderover><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo>.</mo></mrow></mstyle></mrow></math> If (I – M2)N = 2I, then the positive integral value of is

N=M2+M4+.....+M98=(−α2 I)+(−α2 I)2+....+(−α2 I)49=I(−α2+α4−α6+....−α98)N = −I(α2−α4+α6.......+α98)=−Iα2(1−(−α2)49)1−(−α2)N = −I α2(1+α98)1+α2Now (I−m2)N=−2I(I+α2I)(−Iα2⋅(1+α98)1+α2=−2I? α100 + α2 = 2? α = ±1

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Maths NCERT Exemplar Solutions Class 12th Chapter Four Maths Class 12th Maths Matrices

Let $M = [\begin{matrix} 0 & - α \\ α & 0 \end{matrix}],$ where is a non-zero real number an $N = \sum_{k = 1}^{49} M^{2 k} .$ If (I – M²)N = 2I, then the positive integral value of is

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Payal Gupta | Contributor-Level 10

2 months ago

$N = M^{2} + M^{4} + . . . . . + M^{98}$

$= (- α^{2} I) + {(- α^{2} I)}^{2} + . . . . + {(- α^{2} I)}^{49}$

$= I (- α^{2} + α^{4} - α^{6} + . . . . - α^{98})$

N = $- I (α^{2} - α^{4} + α^{6} . . . . . . . + α^{98})$

$= - I \frac{α^{2} (1 - {(- α^{2})}^{49})}{1 - (- α^{2})}$

N = $\frac{- I α^{2} (1 + α^{98})}{1 + α^{2}}$

Now $(I - m^{2}) N = - 2 I$

$(I + α^{2} I) \frac{(- I α^{2} \cdot (1 + α^{98})}{1 + α^{2}} = - 2 I$

? α¹⁰⁰ + α² = 2

? α = $\pm 1$

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Let $M = [\begin{matrix} 0 & - α \\ α & 0 \end{matrix}],$ where is a non-zero real number an $N = \sum_{k = 1}^{49} M^{2 k} .$ If (I – M²)N = 2I, then the positive integral value of is

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Let M=[0−αα0], where is a non-zero real number an N=∑k=149M2k. If (I – M2)N = 2I, then the positive integral value of is

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Let $M = [\begin{matrix} 0 & - α \\ α & 0 \end{matrix}],$ where is a non-zero real number an $N = \sum_{k = 1}^{49} M^{2 k} .$ If (I – M²)N = 2I, then the positive integral value of is