### Composition and Matrix Multiplication

Remark The following exercises show that, for the matrix product, AB = O does not necessarily imply A = O or B = O.

Exercise For each of the following matrix A, find all the two column matrices X satisfying AX = O (the zero matrix).

1)

 A = [ 1 2 ]

Answer The matrix X has to be 2 by 2. Let

 X = [ x y ] z w

Then

 AX = [ x + 2z y + 2w ]

so that AX = O means

x + 2z = 0,
y + 2w = 0.

This further means x = -2z, y = -2w, with z and w arbitrary. We conclude that

 X = [ - 2z - 2w ] = z[ -2 0 ] + w[ 0 -2 ] z w 1 0 0 1

2)

 A = [ 1 ] 2

Answer The matrix X has to be 1 by 2. Let X = [x y]. Then

 AX = [ x y ] 2x 2y

so that AX = O means

x = 0, 2x = 0,
y = 0, 2y = 0.

We conclude that X = O = [0 0].

3)

 A = [ 1 1 ] 1 -1

Answer The matrix X has to be 2 by 2. Let

 X = [ x y ] z w

Then

 AX = [ x + z y + w ] x - z y - w

so that AX = O means

x + z = 0, x - z = 0,
y + w = 0, y - w = 0.

The first two equations tell us x = z = 0. The last two equations tell us y = w = 0. We conclude that AX = O implies X = O.

4)

 A = [ 2 4 ] -1 -2

Answer The matrix X has to be 2 by 2. Let X be as before. Then

 AX = [ 2x + 4z 2y + 4w ] - x - 2z - y - 2w

so that AX = O means

2x + 4z = 0, - x - 2z = 0,
2y + 4w = 0, - y - 2w = 0.

The first two equations tell us x = - 2z, z arbitrary. The last two equations tell us y = - 2w, w arbitrary. We conclude that

 X = [ - 2z - 2w ] = z[ -2 0 ] + w[ 0 -2 ] z w 1 0 0 1

5)

 A = [ 0 0 ] 0 0

Answer For any 2 by 2 matrix X, we always have AX = O.

6)

 A = [ 1 -2 3 ] -2 3 -4

Answer The matrix X has to be 3 by 2. Let

 X = [ x y ] z w u v

Then AX = O means

x - 2z + 3u = 0, - 2x + 3z - 4u = 0,
y - 2w + 3v = 0, - 2y + 3w - 4v = 0.

The first two equations tell us x = u, z = 2u, u arbitrary. The last two equations tell us y = v, w = 2v, v arbitrary. We conclude that

 X = [ u v ] = u[ 1 0 ] + v[ 0 1 ] 2u 2v 2 0 0 2 u v 1 0 0 1

7)

 A = [ 1 -2 ] -2 3 3 -4

Answer The matrix X has to be 2 by 2. Let

 X = [ x y ] z w

Then AX = O means

x - 2z = 0, - 2x + 3z = 0, 3x - 4z = 0,
y - 2w = 0, - 2y + 3w = 0, 3y - 4w = 0.

From the equations, we conclude that X has to be the zero matrix.

Remark Here are some further problems for you to think about.

1. Let X = [x y]. Then AX = [Ax Ay] by this formula. Thus AX = O means solving Ax = 0 and Ay = 0. More precisely, we solve the homogeneous system Ax = 0 and put two independent solutions as the two columns of X. Check out the discussion with the concrete examples we worked out above.
2. When can we find nonzero X so that AX = 0? Answer the question in terms of properties of the systems Ax = b.
3. What if X has 3 columns? four columns?
4. For a given matrix A, find X such that XA = O. First try the problem with the matrices given in the exercise.