Linear projection/Idempotent/22/Determine/Exercise

a) Show that the ${}2\times 2$-matrices

${\begin{pmatrix}{\frac {1\pm {\sqrt {1-4bc}}}{2}}&b\\c&{\frac {1\mp {\sqrt {1-4bc}}}{2}}\end{pmatrix}}$

describe projections. Here, ${}b,c\in K$ are such that a square root ${}{\sqrt {1-4bc}}$ exists.

b) Determine all ${}2\times 2$-matrices

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

that describe a projection.

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