Question

Given \(A = \left( {\begin{array}{*{20}{c}} 2&5\\ 0&3 \end{array}} \right)\). The value of the determinant |A4 -5A3 + 6A2 + 2I| = ______

Answer 1

Concept:

Cayley – Hamilton theorem

It states that a matrix satisfies its own characteristic equation.

That is, if the characteristic equation of an n × n matrix A is λⁿ + a_n −1 λⁿ⁻¹ + … + a₁ λ + a₀ = 0, then

Aⁿ + a_n−1Aⁿ⁻¹ + ⋯ + a₁A + a₀I = 0.

Calculation:

The characteristic equation for given matrix is:

|A - λI | = 0

\(\left[ {\begin{array}{*{20}{c}} 2&5\\ 0&3 \end{array}} \right] - λ \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {2 - λ }&5\\ 0&{3 - λ } \end{array}} \right]\)

\(\left| {\left[ {\begin{array}{*{20}{c}} {2 - λ }&5\\ 0&{3 - λ } \end{array}} \right]} \right| = 0\)

(λ - 2)(λ - 3) = 0

λ² - 5λ + 6 = 0

The characteristic equation is:

A² - 5A + 6I = 0

|A4 -5A3 + 6A2 + 2I| = |A²(A² - 5A + 6) + 2I| 

| 0 + 2I |

\(\left| {\left[ {\begin{array}{*{20}{c}} 2&0\\ 0&2 \end{array}} \right]} \right| = 4\)