Given the matrix
P=beginbmatrix1&2\2&3endbmatrix,
we want to find P2−4P−I, where I is the identity matrix:
I=beginbmatrix1&0\0&1endbmatrix.
Calculate {P2}
P2=P×P=beginbmatrix1&2\2&3endbmatrix×beginbmatrix1&2\2&3endbmatrix=beginbmatrix5&8\8&13endbmatrix.
Calculate {4P}
4P=4×beginbmatrix1&2\2&3endbmatrix=beginbmatrix4&8\8&12endbmatrix
Calculate {P2−4P}
P2−4P=beginbmatrix5&8\8&13endbmatrix−beginbmatrix4&8\8&12endbmatrix=beginbmatrix5−4&8−8\8−8&13−12endbmatrix=beginbmatrix1&0\0&1endbmatrix
Calculate {P2−4P−I}
P2−4P−I=beginbmatrix1&0\0&1endbmatrix−beginbmatrix1&0\0&1endbmatrix=beginbmatrix0&0\0&0endbmatrix
Thus, the result of P2−4P−I is
beginbmatrix0&0\0&0endbmatrix