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Question
I don't understad what to do for the queston, Thanks for the help.

Let

A =    a b
      c d

be a 2x2 matrix. Let vector v = <a,b> and vector w = <c,d> be the column vectors of A.

(a) Compute the cross product of vector v_R3 = <a,b,0> and vector w_R3 = <c,d,0>
(b) Show that |det(A)| is the area of the parallelogram of R^2 with sides vector v and vector w.
(c) Conclude that A is invertible exactly when its columns vector v and vector w are not parallel.

Answer
a) cross-product is k(ac-bd), k = unit vector, and can be written |v||w|sin(theta) where theta is the angle between the vectors and this expression = area of parallelogram formed by v and w.

b) det(A) = ac-bd, which is the magnitude of the cross-product and thus the area of the parallelogram

c) from a), the cross-product is 0 when v and w are parallel which is when theta = 0. Cross-product = 0 means det(A) = 0 which means A is not invertible.

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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