# 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...

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.

*print*Print*list*Cite

### 2 Answers

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>` .**

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.