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Let A = [[a, b],[c, d]]be a 2x2 matrix. Let vector v = <a,b> and vector w =...

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mrbest55 | Student | Salutatorian

Posted April 8, 2013 at 6:52 AM via web

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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.

Conclude that A is invertible exactly when its columns vector v and vector w are not parallel.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted April 8, 2013 at 3:32 PM (Answer #1)

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You need to use the equation that relates two parallel vectors, such that:

`bar v || bar w <=> a/c = b/d => ad = bc => ad - bc = 0`

You need to prove that matrix A is invertible if the vectors are not parallel.

You should remember that a matrix is invertible if `det A != 0` , hence, evaluating the determinant of matrix A, yields:

`det A = [(a,c),(b,d)] ` `=> det A = ad - bc`

You need to notice that `det A != 0` if `ad - bc != 0` , hence, since `ad - bc != 0` , then the vector `bar v = <a,b>` is not parallel to `bar w = <c,d>` .

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pramodpandey | College Teacher | Valedictorian

Posted April 11, 2013 at 5:06 AM (Answer #1)

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We wish to prove that  A is invertible ,if  columns vector v and vector w are not parallel i.e cross product of  vector v and vector w  does not vanish.

`v xxw =|[a,b],[c,d]|`

`` `=ad-bc`

If `ad-bc!=0`  ,then vector v and w can not be parallel.

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