In textbook Elementary Linear Algebra (Howard Anton) onSection 1.4, Question 14,

Question

In textbook Elementary Linear Algebra (Howard Anton) onSection 1.4, Question 14, the solution says to multiply both sidesof the polynomial by the inverse of A. I am not sure why thisstep is done? How would I kow to multiply both sides byA-1, what is the reason for multiplying by thisvalue? The question is: Show that if a square matrix Asatisfies A2-3A+I = 0 then A-1=3I-A Thanks Sabrina In textbook Elementary Linear Algebra (Howard Anton) onSection 1.4, Question 14, the solution says to multiply both sidesof the polynomial by the inverse of A. I am not sure why thisstep is done? How would I kow to multiply both sides byA-1, what is the reason for multiplying by thisvalue? The question is: Show that if a square matrix Asatisfies A2-3A+I = 0 then A-1=3I-A Thanks Sabrina

Explanation / Answer

given that the square matrix A satisfies the equationA2 - 3A + I = 0 .......... (1) we need to show that A-1 = 3I - A from (1), we have I = 3A - A2 observe the power of A in this is 2, AA-1 = I =A-1A, IA-1 = A-1 =A-1I multiply with A-1 on both sides of I = 3A -A2 from left, we have A-1I = 3A-1A -A-1A2 thus A-1 = 3I - A thus A-1 = 3I - A

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