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# Linear Algebra: Rank

10. Let A = a11 a12 a13 Show that A has rank 2 if and only if one or more of
a21 a22 a23 the determinants

a11 a12 a11 a13 a12 a13
a21 a22 a21 a23 a22 a23 are non zero.

14. Use the result in Exercise 10 to show that the set of points (x, y, z) in R3 for which the matrix
x y z has rank 1 is the curve with parametric equations x = t, y = t2, z = t3
1 x y

15. Prove: If k does not equal to zero, then kA have the same rank.

See attached file for full problem description.

#### Solution Preview

10. Let A = a11 a12 a13 Show that A has rank 2 if and only if one or more of
a21 a22 a23 the determinants

a11 a12 a11 a13 a12 a13
a21 a22 a21 a23 a22 a23 are non zero.

The rank of a matrix A is the number of linear independent rows (or columns) in A. For a m x n matrix, the rank of A is at most the minimum of n and m.

In this problem, A is a 2x3 matrix, so the rank of A is either 1 or 2 (only the zero matrix can have rank 0, and we know that the rank of A is at most 2).

Two vectors are linearly independent if and only if the determinant of their matrix is non zero.

Now, let's prove the statement...

(1) A has rank 2 if at least one of the determinants is not 0.

Assume that the determinants of one of the matrices is non-zero. That means that the vectors that form the columns of that matrix are linearly independent. (So, if the first matrix a11 a12
a21 a22
has a non-zero determinant, ...

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