Section 18 Matrix Multiplication

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Here we will practice multiplying matrices in R. First, let’s define a few matrices.I’m using a trick here. By putting the assignment in parentheses, it assigns the matrix and displays it.s

(A = cbind(c(1,2,3),c(4,5,6),c(1,1,-1)))

##      [,1] [,2] [,3]
## [1,]    1    4    1
## [2,]    2    5    1
## [3,]    3    6   -1

(B = cbind(c(1,-1,1),c(1,1,1),c(0,2,1)))

##      [,1] [,2] [,3]
## [1,]    1    1    0
## [2,]   -1    1    2
## [3,]    1    1    1

(C = cbind(c(2,1,1),c(1,0,1),c(1,-3,1),c(3,2,1)))

##      [,1] [,2] [,3] [,4]
## [1,]    2    1    1    3
## [2,]    1    0   -3    2
## [3,]    1    1    1    1

We multiply using the %*% command. As seen here:

A %*% B

##      [,1] [,2] [,3]
## [1,]   -2    6    9
## [2,]   -2    8   11
## [3,]   -4    8   11

Note that

B %*% A

##      [,1] [,2] [,3]
## [1,]    3    9    2
## [2,]    7   13   -2
## [3,]    6   15    1

1. What do these last two multiplications say about the matrix product AB and BA? This is a very important property (or, perhaps, lack of property) of matrix multiplication.

2. Try multiplying BC and CB. What happens? And why?

3. The transpose of a matrix is computed by t(A). Compute the transpose of the matrices A, B, C and be sure that you all understand what it does.

4. The command diag(n) gives the n x n identity matrix. This is denoted \(I_n\). For example, here is \(I_3\).

diag(3)

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

Compute \(I_2\), \(I_4\), and \(I_5\) and be sure you all agree on what the identity matrix is.

5. Multiply the matrices A and B by the appropriately-sized identity matrix. Multiply both ways, A I and I A, and agree upon what multiplying by the identity does.

6. Multiply C by an identity matrix I C and C I. You might need a different size on the left and on the right.

7. Our topic for Thursday (tomorrow) is the inverse of a matrix. You compute the inverse of the matrix A with solve(A). Try this

solve(B)

##      [,1] [,2] [,3]
## [1,] -0.5 -0.5    1
## [2,]  1.5  0.5   -1
## [3,] -1.0  0.0    1

B %*% solve(B)

##      [,1] [,2] [,3]
## [1,]    1    0    0
## [2,]    0    1    0
## [3,]    0    0    1

8. Multiply A by its inverse and look closely at the answer you get.

solve(A)

##            [,1]       [,2]       [,3]
## [1,] -1.8333333  1.6666667 -0.1666667
## [2,]  0.8333333 -0.6666667  0.1666667
## [3,] -0.5000000  1.0000000 -0.5000000

A %*% solve(A)

##              [,1]          [,2]          [,3]
## [1,] 1.000000e+00  0.000000e+00  0.000000e+00
## [2,] 3.330669e-16  1.000000e+00 -1.110223e-16
## [3,] 1.110223e-16 -2.220446e-16  1.000000e+00

9. Some matrices do not have inverses. Try computing the inverse of the following matrices. We will discuss this tomorrow!

(M1 = cbind(c(3,5),c(-2,1)))

##      [,1] [,2]
## [1,]    3   -2
## [2,]    5    1

(M2 = cbind(c(4,3),c(5,4)))

##      [,1] [,2]
## [1,]    4    5
## [2,]    3    4

(M3 = cbind(c(4,2),c(10,5)))

##      [,1] [,2]
## [1,]    4   10
## [2,]    2    5

10. Enter the matrix A in problem 3.7 in the homework. Then compute the matrix G which is A times its transpose. Discuss its meaning.