Inverses of Matrices

If A is a p × q matrix, by the previous exercise,

A = (Ae1 Ae2 . . . Aeq) .

In particular, when we multiply matrices

AB = (ABe1 ABe2 . . . ABeq)

(and if either side of this equation is defined, then both sides are and they are equal). In other words, the columns of AB are A times the columns of B.

This next exercise is particularly vital:

3.9 If A is a matrix, and x a vector, prove that Ax is a sum of the columns of A, each weighted by entries of x:

Ax = x1 (Ae1) + x2 (Ae2) + · · · + xn (Aen) .

Review problems

3.10 True or false (if false, give a counterexample):

a.If the second column of B is 3 times the first column, then the same is true of AB.

b.Same question for rows instead of columns.

3.11Can you find matrices A and B so that A is 3 × 5 and B is 5 × 3, and

AB = I?

3.12Prove that the rows of AB are the rows of A multiplied by B.

3.13The Fibonacci numbers are the numbers x0 = 1, x1 = 1, xn+1 = xn +xn−1. Write down x0, x1, x2, x3 and x4. Let

(a) The

original

Figure 3.1: Faces formed by y = Ax. Each face is centered at the origin.

For each matrix M = A, B, C, D, E, F draw Mx and My (in a di erent colour for each matrix), and explain in words what each matrix is “doing” (for example, rotating, flattening onto a line, expanding, contracting, etc.).

3.15 The first picture in figure 3.1 is the original in the x1, x2 plane, and the center of the circular face is at the origin. If we pick a matrix A and set y = Ax, and draw the image in the y1, y2 plane, which matrix below draws which picture?

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3.16Can you figure out which matrices give rise to the pictures in the last problem, just by looking at the pictures? Assume that you known that all the entries of each matrix are integers between -2 and 2.

3.17What are the simplest examples you can find of 2×2 matrices A for which taking vectors x to Ax

(a) contracts the plane,

(b) dilates the plane,

(c) dilates one direction, while contracting another,

(d) rotates the plane by a right angle,

(e) reflects the plane in a line,

(f) moves the vertical axis, but leaves every point of the horizontal axis where

it is (a “shear”)?

Inverses

A matrix is square if it has the same number of rows as columns. If A is a square matrix, an inverse is a square matrix B of the same size as A so that

AB = BA = I.

3.18 If A, B and C are square matrices, and AB = I and CA = I, prove that B = C. In particular, there is only one inverse (if there is one at all).

So we can unambiguously write the inverse of A (if there is one) as A−1.

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3.20Which 1 × 1 matrices have inverses, and what are their inverses?

3.21By multiplying out the matrices, prove that any 2 × 2 matrix

as long as ad − bc 6= 0.

3.22If A and B are invertible matrices, prove that AB is invertible, and

(AB)−1 = B−1A−1.

3.23Prove that A−1 −1 = A, for any invertible square matrix A.

3.24If A is invertible, prove that Ax = 0 only for x = 0.

3.25If A is invertible, and AB = I, prove that A = B−1 and B = A−1.

Review problems

3.26Write down a pair of nonzero 2 × 2 matrices A and B for which AB = 0.

3.27If A is an invertible matrix, prove that Ax = Ay just when x = y.

3.28If a matrix M splits up into square blocks like

!

explain how to find M−1 in terms of A−1 and D−1. (Warning: for a matrix which splits into blocks like

!

the inverse of M cannot be expressed in any elementary way in terms of the blocks and their inverses.)

Original

Figure 3.2: Images coming from some matrices, and from their inverses.

3.29 Figure 3.2 shows how various matrices (on the left hand side) and their inverses (on the right hand side) a ect vectors. But the two columns are scrambled up. Which right hand side picture is produced by the inverse matrix of each left hand side picture?

Elimination by Matrix Multiplication

Take the 3 × 3 identity matrix I, and swap the first two rows. Call the resulting matrix A:

It turns out that, for any vector x, the vector Ax is just the vector x with the first two rows swapped. Why? First, let’s see if this is true for the example of x = e1. We know that Ae1 is the first column of A. So Ae1 is the first column

of I, but with the first two rows swapped. The first column of I is e1. So Ae1 is e1 with the first two rows swapped. The same reasoning exactly works with e1 replaced by e2 or e3. So if x = e1 or x = e2 or x = e3, then Ax is x with the first two rows swapped. Since we can write any vector as x = x1e1 +x2e2 +x3e3, it is enough the check what happens if we take x = e1 and then check x = e2 and then check x = e3, as we have done. So for any vector x, the vector Ax must be just x with the first two rows swapped. A row operation is the process of adding a multiple of a row to another row, swapping two row, or rescaling a row. The same reasoning works exactly if we start with the n × n identity matrix I and let A be the result of carrying out any of the row operations that we came across in elimination. Let’s make that more precise and summarize what we have learned.

Lemma 3.1. Carry out some row operation on I, or more generally you can carry out several row operations to I, as many as you like. Call the resulting matrix A. Then for any vector x, the vector Ax is the result of carrying out exactly those same row operations on x.

For example, if we start with the 3 × 3 identity matrix I, and add 7 row 1 to row 3 then

Our lemma claims that Ax is just x with 7 row 1 added to row 3. Let’s check:

Corollary 3.2. Carry carry out several row operations on I, as many as you like. Call the resulting matrix A. Then for any matrix B, the matrix AB is just the result of applying exactly those same row operations to B.

Proof. The columns of AB are A times the columns of B.

Corollary 3.3. If A is the matrix you get from I by carrying out some row operations, and B is the matrix that you get by carrying out some other row operations, then AB is the matrix that you get from I by carrying out first the row operations that gave you B, and then those that gave you A.

Proof. The matrix A acts on a vector x by carrying out those row operations that gave us A from I, and the same is true for B. But (AB)x = A(Bx), so AB is the matrix that carries out the row operations first of B and then of A. So then AB = ABI is the matrix you get by carrying out the row operations of B on I, and then the row operations of A.

Corollary 3.4. If A is the matrix that you get from I by carrying out some row operations, then the matrix A−1 is the matrix that you get from I by carrying out the inverse row operations, in the reverse order.

Proof. Let B be the matrix that you get from I by carrying out the inverse row operations, in the reverse order. Then AB = I and BA = I, doing and then underdoing various operations.