26. If <math><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><mtable><mtr columnalign="center"><mtd columnalign="center"><mn>1</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>2</mn></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>2</mn></mtd><mtd columnalign="center"><mn>1</mn></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mn>2</mn></mtd><mtd columnalign="center"><mn>0</mn></mtd><mtd columnalign="center"><mn>3</mn></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow></math> prove that <math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>6</mn><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>7</mn><mi>A</mi><mo>+</mo><mn>2</mn><mi>I</mi><mo>=</mo><mn>0</mn></mrow></math>

Given A = [102021203]A2 = AA = [102021203][102021203]=[1×1+0×0+2×21×0+2×0+2×01×2+0×1+2×30×1+2×0+1×2a×0+2×2+1×00×2+2×1+1×32×1+0×0+3×22×0+0×2+3×02×2+0×1+3×3]=[1+4 0 2+62 4 2+32+6 0 4+9]=[5082458013]A3 = A2A = [5082458013][102021203]=[5×1+0×0+8×25×0+0×2+8×05×2+0×1+8×32×1+4×0+5×22×0+4×2+5×02×2+4×1+5×38×1+0×0+13×28×0+0×2+0×08×2+0×1+13×3][5+16010+242+1084+4+158+26016+39] =[210341282334055]So, A3-6A2 + 7A + 2I=[210341282334055]−6[5082458013]+7[102021203]+2[100010001]=[210341282334055]−[3004812243048078]+[7014014714021]+[200020002]=[21−30+7+20−0+0+034−48+14+012−12+0+08−24+14+223−30+7+034−48+14+00−0+0+055−78+21+2]=[000000000]=0 (2000 matrix )Hence proud

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26. If $A = [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$ prove that $A^{3} - 6 A^{2} + 7 A + 2 I = 0$

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Vishal Baghel | Contributor-Level 10

4 months ago

Given A = $[\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$

A² = AA = $[\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}] [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$

$= [\begin{matrix} 1 \times 1 + 0 \times 0 + 2 \times 2 & 1 \times 0 + 2 \times 0 + 2 \times 0 & 1 \times 2 + 0 \times 1 + 2 \times 3 \\ 0 \times 1 + 2 \times 0 + 1 \times 2 & a \times 0 + 2 \times 2 + 1 \times 0 & 0 \times 2 + 2 \times 1 + 1 \times 3 \\ 2 \times 1 + 0 \times 0 + 3 \times 2 & 2 \times 0 + 0 \times 2 + 3 \times 0 & 2 \times 2 + 0 \times 1 + 3 \times 3 \end{matrix}]$

$= [\begin{array}{l} 1 + 4 0 2 + 6 \\ 2 4 2 + 3 \\ 2 + 6 0 4 + 9 \end{array}] = [\begin{matrix} 5 & 0 & 8 \\ 2 & 4 & 5 \\ 8 & 0 & 13 \end{matrix}]$

A³ = A²A = $[\begin{matrix} 5 & 0 & 8 \\ 2 & 4 & 5 \\ 8 & 0 & 13 \end{matrix}] [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$

$= [\begin{matrix} 5 \times 1 + 0 \times 0 + 8 \times 2 & 5 \times 0 + 0 \times 2 + 8 \times 0 & 5 \times 2 + 0 \times 1 + 8 \times 3 \\ 2 \times 1 + 4 \times 0 + 5 \times 2 & 2 \times 0 + 4 \times 2 + 5 \times 0 & 2 \times 2 + 4 \times 1 + 5 \times 3 \\ 8 \times 1 + 0 \times 0 + 13 \times 2 & 8 \times 0 + 0 \times 2 + 0 \times 0 & 8 \times 2 + 0 \times 1 + 13 \times 3 \end{matrix}]$

$[\begin{matrix} 5 + 16 & 0 & 10 + 24 \\ 2 + 10 & 8 & 4 + 4 + 15 \\ 8 + 26 & 0 & 16 + 39 \end{matrix}]$ $= [\begin{array}{l} 21 & 0 & 34 \\ 12 & 8 & 23 \\ 34 & 0 & 55 \end{array}]$

So, A³-6A² + 7A + 2I

$= [\begin{array}{l} 21 & 0 & 34 \\ 12 & 8 & 23 \\ 34 & 0 & 55 \end{array}] - 6 [\begin{array}{l} 5 & 0 & 8 \\ 2 & 4 & 5 \\ 8 & 0 & 13 \end{array}] + 7 [\begin{array}{l} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{array}] + 2 [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

$= [\begin{array}{l} 21 & 0 & 34 \\ 12 & 8 & 23 \\ 34 & 0 & 55 \end{array}] - [\begin{matrix} 30 & 0 & 48 \\ 12 & 24 & 30 \\ 48 & 0 & 78 \end{matrix}] + [\begin{matrix} 7 & 0 & 14 \\ 0 & 14 & 7 \\ 14 & 0 & 21 \end{matrix}] + [\begin{array}{l} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}]$

$= [\begin{array}{l} 21 - 30 + 7 + 2 & 0 - 0 + 0 + 0 & 34 - 48 + 14 + 0 \\ 12 - 12 + 0 + 0 & 8 - 24 + 14 + 2 & 23 - 30 + 7 + 0 \\ 34 - 48 + 14 + 0 & 0 - 0 + 0 + 0 & 55 - 78 + 21 + 2 \end{array}]$

$= [\begin{array}{l} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] = 0$ (2000 matrix )

Hence proud

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Let $A = [\begin{matrix} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{matrix}]$

Given $A = [\begin{matrix} 1 & 0 & 1 \end{matrix}] = [\begin{matrix} 2 & 0 & 2 \end{matrix}]$ ...(1)

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$[\begin{matrix} - x_{1} + z_{1} & - x_{2} + z_{2} & - x_{3} + z_{3} \end{matrix}] = [\begin{matrix} 4 & 0 & 4 \end{matrix}]$

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26. If $A = [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$ prove that $A^{3} - 6 A^{2} + 7 A + 2 I = 0$

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26. If A=[102021203] prove that A3−6A2+7A+2I=0

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26. If $A = [\begin{matrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{matrix}]$ prove that $A^{3} - 6 A^{2} + 7 A + 2 I = 0$