Suppose A is any 3\[ \times \] 3 non-singular matrix and ( A - 3I) (A - 5I) = O where I = I3 and O = O3. If \[\alpha \]A + \[\beta \]A-1 = 4I, then \[\alpha \] + \[\beta \] is equal to :
Solutions · 1
Official solution
Given,
( A - 3I) (A - 5I) = O
\[ \Rightarrow \] A2 - 8A + 15I = O
Multiplying both sides by A- 1, we get,
A- 1A.A - 8A- 1A + 15A- 1I = A- 1O
\[ \Rightarrow \] A - 8I + 15A- 1 = O
\[ \Rightarrow \] A + 15A- 1 = 8I
\[ \Rightarrow \] 2A+215A−1=4IComparing with the equationαA +βA-1 = 4I, we getα=21andβ=215∴α+β=21+215=216