1 votes 1 votes Given that a matrix $A_{3\times3},$which is not idempotent matrix.And $A^{3}=A.$ Then find them, $1)$ Eigen Values $2)$ Trace of the matrix$=$Sum of Leading Diagonal Elements$=\sum a_{ij},$ where $i=j$ $3)Det(A)$ Linear Algebra engineering-mathematics linear-algebra eigen-value + – Lakshman Bhaiya asked Oct 25, 2018 edited Oct 25, 2018 by Lakshman Bhaiya Lakshman Bhaiya 472 views answer comment Share Follow See all 4 Comments See all 4 4 Comments reply minal commented Oct 25, 2018 reply Follow Share is it correct A^3= A A^3-A=0 A(A^2-I)=0 A(A+I)(A-I)=0 A= 0, A=I OR A=-I am going in right way ? 0 votes 0 votes Lakshman Bhaiya commented Oct 25, 2018 i edited by Lakshman Bhaiya Oct 25, 2018 reply Follow Share $A^{3}=A$ $|A^{3}|=|A|$--->(1) for eigen values:Characteristic equation $|A-\lambda I|=0$ $\Rightarrow|A|=|\lambda I|$ $\Rightarrow|A|=|\lambda |$ Cayley–Hamilton Theorem: A square matrix satisfies its own characteristic equation. Put the value in the equation $(1),$ $|\lambda|^{3}=|\lambda|$ $\lambda^{3}-\lambda=0$ $\lambda(\lambda^{2}-1=0$ $\lambda(\lambda+1).(\lambda-1)=0$ so $,\lambda=0,-1,1$ 3 votes 3 votes hitendra singh commented Oct 25, 2018 reply Follow Share If A3=A then what do you mean by not idempotent matrix??? 0 votes 0 votes Lakshman Bhaiya commented Oct 25, 2018 reply Follow Share $A^{2}\neq A$ 1 votes 1 votes Please log in or register to add a comment.