Question

If \(A = \left[ {\begin{array}{*{20}{c}} 4&2\\ { - 1}&1 \end{array}} \right]\) then (A – 2I) (A – 3I) is
1. A
2. I
3. 0
4. 5 I

Answer 1

Correct Answer - Option 3 : 0

\(A = \left[ {\begin{array}{*{20}{c}} 4&2\\ { - 1}&1 \end{array}} \right]\)

(A – 2I)(A – 3I)

= A² – 3A – 2A + 6I

= A² – 5A + 6        ---(1)

\(\left| {A - \lambda I} \right| = \left[ {\begin{array}{*{20}{c}} {4 - \lambda }&2\\ { - 1}&{1 - \lambda } \end{array}} \right]\)

(4 - λ)(1 - λ) + 2 = 0

4 – 5λ + λ² + 2 = 0

λ² – 5λ + 6 = 0

Now,

Cay ley – Hamilton theorem says that a matrix A satisfies it characteristic equation:

∴ A² – 5A + 6 = 0