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Question:

Find the adjoint of the matrix A = βŽ›βŽœβŽπ•’΅π•’Άπ•’Ά21π•’Ά2π•’Ά1⎞⎟⎠ and hence show that A.(adjA) = |A|I3

Solution:

|A| = βˆ£βˆ£βˆ£βˆ£π•’΅π•’Άπ•’Ά21π•’Ά2π•’Ά1∣∣∣∣ = 27
Adjoint of a matrix is the transpose of its' cofactor matrix.
adj(A) = βŽ‘βŽ’βŽ£οΏ½π•’Ί3π•’Ίπ•’Ίπ•’Ί3⎀βŽ₯⎦
A.adj(A) = βŽ‘βŽ’βŽ£π•’΅π•’Άπ•’Ά21π•’Ά2π•’Ά1⎀βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ£οΏ½π•’Ί3π•’Ίπ•’Ίπ•’Ί3⎀βŽ₯⎦ = ⎑⎒⎣270002700027⎀βŽ₯⎦ = 27⎑⎒⎣100010001⎀βŽ₯⎦
Hence proved.