worksheet12 - Math 1300 Winter 2017 Labs B05\u2013B08 Worksheet#12 1 2 0 1 \u0012 \u0013 \u22122 0 0 3 1 0 0 1 0 0 3 2 1 3 0 C = 0 3 1 X = 1 Let A =,B= 0 0 2 1 2 3

# worksheet12 - Math 1300 Winter 2017 Labs B05u2013B08...

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Math 1300 Winter 2017 Labs B05–B08 Worksheet #12 1. Let A = 3 2 2 3 , B = - 2 0 0 1 3 0 0 1 2 , C = 3 1 0 0 3 1 0 0 3 , X = 1 2 0 1 0 1 0 0 0 0 2 1 0 0 0 2 . For each matrix find all eigenvalues and corresponding eigenvectors. If the matrix is diag-onalizable, do so (by producing appropriate matricesP, D). If not, explain why not.2. In class we found the eigenvalues and eigenvectors of the matrixB=0-165.(a) FindB2andB3(directly). Mentally imagine (but don’t do) the work of findingB10.(b) DiagonalizeBby finding appropriate matricesPandD.(c) IfP-1BP=Dthen solving forBgivesB=PDP-1.Show that(PDP-1)2=PD2P-1and more generally(PDP-1)n=PDnP-1,nZ+.(d) Use parts (b) and (c) to findB10. Do not expand high powers of numbers like310, butdo any obvious elementary simplifications3. Verify thatA=073-4is diagonalized byP=713-1, and use this information tofind an explicit expression forA10.4. LetA=211010002,B=210011002andC=200111002.(a) Find all eigenvalues and eigenvectors forA,BandC.(b) One ofA, B, Ccan be diagonalized and two cannot. Which? and Why?(c) Diagonalize the one which can be diagonalized.5. Evaluatef(A), whereA=110011001andf(x) = 2 +x2-x3(See question 13 of worksheet 11)6. Set up and solve a system of equations in variablesa, b, cto find a quadratic polynomialp(x) =ax2+bx+cwhose graph passes through points(1,1),(2,-1)and(3,2).7. Find a cubic equation of the formy=ax3+bx+cthat passes through pointsA(-2,-5),B(1,-2),C(2,7).8. WriteA=00-1013222as a product of elementary matrices.9. Find the inverse of3254by using the Matrix Inversion Algorithm.10. Find the distance inR2fromP(1,2)to line3x+ 7y= 13and the distance inR3fromQ(2,0,2)