The value of | U

Engineering

Mathematics

Introduction to Determinants

Question

The value of | U | equals

– 1

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Aⁿ – A^n–2 = A² – I ⇒ A⁵⁰ = A⁴⁸ + A² – I

Further,

A⁴⁸ = A⁴⁶ + A² – I

A⁴⁶ = A⁴⁴ + A² – I

A⁴ = A² + A² – I

________________

A⁵⁰ = 25A² – 24I

Here,

A^{2} = [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{array}]

\Rightarrow A^{50} = [\begin{array}{ccc} 25 & 0 & 0 \\ 25 & 25 & 0 \\ 25 & 0 & 25 \end{array}] - 24 [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1 \end{array}]

∴ |A⁵⁰| = 1

Also tr(A⁵⁰) = 1 + 1 + 1 = 3 further,

[\begin{array}{ccc} 1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1 \end{array}] [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 1 \\ 25 \\ 25 \end{array}] \Rightarrow [\begin{array}{c} x \\ y \\ z \end{array}] \cup_{2} = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]

similarly

\cup_{2} = [\begin{array}{c} 0 \\ 1 \\ 0 \end{array}]

and

\cup_{3} = [\begin{array}{c} 0 \\ 0 \\ 1 \end{array}] \Rightarrow \cup [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]

, i.e. |U| = 1

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