Let A be an mxn matrix and let B be an nxm matrix. Show that det(I_n, B; A, I_m)

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Prove that the trace of a matrix A is equal to the sum of its eigenvalues? School Subjects Questions Answers.com > Wiki Answers > Categories > Jobs & Education > Education > School Subjects > Math and Arithmetic > Proofs View Slide Show Best Answer Given a matrix A=([a,b],[c,d]), the trace of A is a+d, and the det of A is ad-bc. By using the characteristic equation, and representing the eigenvalues with x, we have the equation x2-(a+d)x+(ad-bc)=0 Which, using the formula for quadratic equations, gives us the eigenvalues as, x1=[(a+d)+v((a+d)2-4(ad-bc))]/2 x2=[(a+d)-v((a+d)2-4(ad-bc))]/2 now by adding the two eigenvalues together we get: x1+x2=(a+d)/2+[v((a+d)2-4(ad-bc))]/2+ (a+d)/2-[v((a+d)2-4(ad-bc))]/2 The square roots cancel each other out being the same value with opposite signs, leaving us with: x1+x2=(a+d)/2+(a+d)/2 x1+x2= 2(a+d)/2 x1+x2=(a+d) x1+x2=trace(A)

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