If A = [31-12], prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2 - Mathematics and Statistics

प्रश्न

If A = $[\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$ , prove that A² – 5A + 7I = 0, where I is unit matrix of order 2

बेरीज

उत्तर

A² = A · A = $[\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}]$

= $[\begin{matrix} 9 - 1 & 3 + 2 \\ - 3 - 2 & - 1 + 4 \end{matrix}]$

= $[\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}]$

∴ A² – 5A + 7I = $[\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] - 5 [\begin{matrix} 3 & 1 \\ - 1 & 2 \end{matrix}] + 7 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

= $[\begin{matrix} 8 & 5 \\ - 5 & 3 \end{matrix}] - [\begin{matrix} 15 & 5 \\ - 5 & 10 \end{matrix}] + [\begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}]$

= $[\begin{matrix} 8 - 15 + 7 & 5 - 5 + 0 \\ - 5 + 5 + 0 & 3 - 10 + 7 \end{matrix}]$

= $[\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

= 0.

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Types of Matrices

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पाठ 4: Determinants and Matrices - Exercise 4.6 [पृष्ठ ९५]

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पाठ 4 Determinants and Matrices
Exercise 4.6 | Q 15 | पृष्ठ ९५

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If A = [31-12], prove that A2 – 5A + 7I = 0, where I is unit matrix of order 2 - Mathematics and Statistics

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