A is any matrix satisfying the property\( { A }^{ 2 } = I\), then the value of the expression \( { A }^{ 3 } + { \left(A+I\right) }^{ 2 } -8A - { I }^{ 2 } - { A }^{ 2 } \) is
Answer & explanation
Correct answer: option 2
Given,
\(A^2 = I\)
The given equation is
\(A^3 + (A + I)^2 - 8A - I^2 - A^2\)
\(= A^2.A + A^2 + I^2 + 2IA - 8A - I^2 - A^2\)
\(= IA + I + I^2 +2IA - 8A - I^2 - I\)
\(= 3IA - 8A\)
\(= 3A - 8A = -5A\)