Question

ANSWER c d e ONLY! No need to answer a and b

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2. Determinant function onM 2 (a) Take A E M2. Consider the mapping volA: R2 x R2 - R, which is given by volA(v1, v2) olvA, 2A), for every v1, 02 E R2. Explain why volA is also a volume form (b) Explain why (use section (c) from question 1 above) there is a scalar α(A) E R such that VOLA-α(A) . voi We denote the scalar a(A) by det(A) and call it the determinant of A (c) Consider the function det : M2R which gives every matrix A E M2 its deter- minant det(A) » Show that if A' M2 is obtained from A M2 by adding a scalar multiple of a row to another row then det(A') Recall that A - det(A) . Show that if A" E M2 is obtained from A E M2 by interchanging two rows of A then det(A') - - det(A) Recall that in this case A'- A) . Show that det ("I1 u12 And apply identity (2) 112 20 U22 0U22 u11 . u22 . (Consider ei (u11 (d) Using the properties you discovered in section (c) above, explain how to compute det(A) for A M2 using Gaussian elimination. (e) Compute the determinant of A in the following cases:

Answer 1

fc) dat (A) du A- ad-cb --(ad-le ing the haww 5 A-A A bd 、A FJ,

To dlo-Unis , y0u jou ada multuju, urant いhun "tru funduk y th. a. l 2a 2b