Justify your answer fully. Show and explain all work for each matrix A.(Ct is transpose of C, C- is inverse of C.)

A = -1 8 4

0 1 -10

0 0 -1

A = -1 -4 0

0 1 0

0 10 -1

(a) determine whether or not there exists an orthogonal matrix C such that (Ct)AC=V where V is real and diagonal. If so find C and V;proceed to the next matrix. If not, explain why not and proceed to part (b).

(b) Determine whether or not there exists an invertible matrix C such that (C-)AC=V where V is real and diagonal. If so, find C and V;proceed to the next matrix. If not, explain why not and proceed to part (c).

(c) determine whether or not there exists an invertible matrix C such that (C-)AC=J where J is in Jordan block form. If so, find J.(Note: One does not need to find C.) If not, explain why not and proceed to next matrix.

A = -1 8 4

0 1 -10

0 0 -1

A = -1 -4 0

0 1 0

0 10 -1

(a) determine whether or not there exists an orthogonal matrix C such that (Ct)AC=V where V is real and diagonal. If so find C and V;proceed to the next matrix. If not, explain why not and proceed to part (b).

(b) Determine whether or not there exists an invertible matrix C such that (C-)AC=V where V is real and diagonal. If so, find C and V;proceed to the next matrix. If not, explain why not and proceed to part (c).

(c) determine whether or not there exists an invertible matrix C such that (C-)AC=J where J is in Jordan block form. If so, find J.(Note: One does not need to find C.) If not, explain why not and proceed to next matrix.

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