MATH 235 Final: MATH 235 UMass Amherst math235_spring08_html solution-final
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Show all your work!: (10 points) the matrices a and b below are row equivalent (you do not need to check this fact). The solution is similar to question 1 in midterm 2: (16 points) consider the matrix a = . 0 (a) show that the characteristic polynomial of a is ( 1)( + 1)( 2). 3 (laplace) expansion along the rst column to get det(a i) = ( + 1)[ 2 3 + 2] = ( 1)( + 1)( 2). 0 (b) find a basis of r3 consisting of eigenvectors of a. Answer: the eigenvalues are the roots of the characteristoc polynomial 1 = The 1-eigenspace: ker(a 1i) = ker(a i) = ker . 0 reducing, we get that the row reduced echelon form of a i is . We see that x3 is a free variable, x1 = x3, x2 = x3, and so the general vector in ker(a i) has the from .
