Let $$P=-302056901401121206014$$ and $$A=27$$ω$$2-1-$$ω$$10-$$ω$$-$$ω$$+1$$ where ω$$=-1+i32$$, and $$I3$$ be the identity matrix of order $$3$$ . If the determinant of the matrix $$P-1AP-I32$$ is αω$$2$$, then the value of α is equal to

Question

Let $$P=-302056901401121206014$$ and $$A=27$$ω$$2-1-$$ω$$10-$$ω$$-$$ω$$+1$$ where ω$$=-1+i32$$, and $$I3$$ be the  identity matrix of order $$3$$ . If the determinant of the matrix $$P-1AP-I32$$ is αω$$2$$, then the  value of α is equal to

Infinity Learn · Accepted Answer

$$P-1AP-I2=P-1AP-P-1P2=P-1(A-I)P2=P-12$$|$$A-I$$|$$2$$|P|$$2$$  $$=$$|$$A-I$$|$$2$$                               A−$$I=17$$ω$$2$$−$$1$$−ω−$$110$$−ω−ω                                                                    $$=17$$ω$$20$$−ω−$$61+$$ω$$20$$−ω−ω                 · $$R2$$→$$R2+R1$$                              $$=$$ωω$$+6+$$ω$$1+$$ω$$2$$                        $$=$$ω$$1+$$ω$$2+$$ω$$+6=6$$ω∴A−$$I2=36$$ω$$2$$

Introduction to Determinants

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Introduction to Determinants

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Let P=-302056901401121206014 and A=27ω2-1-ω10-ω-ω+1 where ω=-1+i32, and I3 be the identity matrix of order 3 . If the determinant of the matrix P-1AP-I32 is αω2, then the value of α is equal to

P-1AP-I2=P-1AP-P-1P2=P-1(A-I)P2=P-12|A-I|2|P|2 =|A-I|2 A−I=17ω2−1−ω−110−ω−ω =17ω20−ω−61+ω20−ω−ω · R2→R2+R1 =ωω+6+ω1+ω2 =ω1+ω2+ω+6=6ω∴A−I2=36ω2

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