csat-suneung 2009 Q24

csat-suneung · South-Korea · csat__math-humanities 4 marks Matrices Matrix Power Computation and Application
A $2 \times 2$ square matrix $A$ has the sum of all components equal to 0 and satisfies
$$A ^ { 2 } + A ^ { 3 } = - 3 A - 3 E$$
Find the sum of all components of the matrix $A ^ { 4 } + A ^ { 5 }$. (Here, $E$ is the identity matrix.) [4 points]
A $2 \times 2$ square matrix $A$ has the sum of all components equal to 0 and satisfies

$$A ^ { 2 } + A ^ { 3 } = - 3 A - 3 E$$

Find the sum of all components of the matrix $A ^ { 4 } + A ^ { 5 }$. (Here, $E$ is the identity matrix.) [4 points]