### Composition and Matrix Multiplication

Exercise Let

 A = [ 2 0 2 1 ] 1 1 -1 0 0 1 -2 1

For each of the following matrix B, find a matrix X such that AX = B.

1)

 B = [ 1 ] -1 -2

Answer X must be a vertical vector of dimension 4. The equation AX = B is the usual system of 3 linear equations in 4 variables. It is in fact this system. By the earlier computation, the augmented matrix [A B] has the following row echelon form.

 [ 1 1 -1 0 -1 ] 0 1 -2 1 -2 0 0 0 3 -1

From this we find one solution

 X = [ -1 ] -2 0 -1/3

2)

 B = [ 1 1 ] -1 0 -2 1

 b1 = [ 1 ],     b2 = [ 1 ] -1 0 -2 1

be the two columns of B. Let x, y be the two columns of X. By AX = A[x y] = [Ax Ay], AX = B is equivalent to two separate systems Ax = b1, Ay = b2. The first system has been studied in the problem above and a solution has been obtained. The second system has an obvious solution

 y = [ 0 ] 0 0 1

Combining the solutions together, we get a solution

 X = [ 1 0 ] -2 0 0 0 -1/3 1

of AX = B.

3)

 B = [ 1 0 0 ] 0 1 0 0 0 1

Answer The columns of B are the standard basis vectors e1, e2, e3. By a similar argument as before, the columns of X are the solutions of the systems Ax = ei. Combining the augmented matrices of the three systems, we get

 [A e1 e2 e3] = [A I] = [ 2 0 2 1 1 0 0 ] 1 1 -1 0 0 1 0 0 1 -2 1 0 0 1

Row operations (e.g., as in this exercise) change the matrix into

 [ 1 1 -1 0 0 1 0 ] 0 1 -2 1 0 0 1 0 0 0 3 1 -2 2

By taking [col 1-5], we get a solution (1/3, -1/3, 0, 1/3) of Ax = e1. By taking [col 1-4] and [col 6], we get a solution (1/3, 2/3, 0, -2/3) of Ax = e2. By taking [col 1-4] and [col 7], we get a solution (-1/3, 1/3, 0, 2/3) of Ax = e3. Putting the three columns together, we get a solution

 X = [ 1/3 1/3 -1/3 ] -1/3 2/3 1/3 0 0 0 1/3 -2/3 2/3

of AX = B.

Remark From this sequence of exercises, you can see the relation between solving equations like AX = B and our usual systems Ax = b of linear equations. Apply your new insight to this exercise and this exercise. Moreover, study the existence/uniqueness/general solution problem for AX = B, just as we have done for systems of linear equations.