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final_review (dragged) 2 - MA 36600 FINAL REVIEW 3 i A is...

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MA 36600 FINAL REVIEW 3 i. A is invertible i.e., there exists B such that A B = B A = I . ii. A is nonsingular i.e., the x = 0 is the only solution to A x = 0 . iii. det A = 0. Say that we are given an m × n matrix where the entries are functions of time t : A ( t ) = a ij ( t ) We define the derivative and (indefinite) integral as those m × n matrices d dt A ( t ) = d dt a ij ( t ) and t A ( τ ) d τ = t a ij ( τ ) d τ . We have the Product Rule i.e., given an m × p matrix A = A ( t ) and a p × n matrix B = B ( t ), d dt A ( t ) B ( t ) = A ( t ) d B dt ( t ) + d A dt ( t ) B ( t ) . We have the Fundamental Theorem of Calculus i.e., given an m × n matrix A = A ( t ), d dt t A ( τ ) d τ = A ( t ) . § 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors. A system of first order linear di ff erential equations can be expressed in the form d dt x 1 x 2 . . . x m m -dim’l vector = p 11 ( t ) p 12 ( t ) · · · p 1 n ( t ) p 21 ( t ) p 22 ( t ) · · · p 2 n ( t ) .
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