csat-suneung 2007 Q21

csat-suneung · South-Korea · csat__math-humanities 3 marks Matrices Linear System and Inverse Existence

For a $2 \times 2$ square matrix $A$ satisfying $( A + E ) ^ { 2 } = A$ and a matrix $\binom { p } { q }$, $$\left( A + A ^ { - 1 } \right) \binom { p } { q } = \binom { 3 } { - 7 }$$ holds. Find the value of $p ^ { 2 } + q ^ { 2 }$. (Note: $E$ is the identity matrix.) [3 points]

For a $2 \times 2$ square matrix $A$ satisfying $( A + E ) ^ { 2 } = A$ and a matrix $\binom { p } { q }$,
$$\left( A + A ^ { - 1 } \right) \binom { p } { q } = \binom { 3 } { - 7 }$$
holds. Find the value of $p ^ { 2 } + q ^ { 2 }$.\\
(Note: $E$ is the identity matrix.) [3 points]

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