Matrix multiplication (%*%) in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of matrix multiplication with compatible dimensions

What is the output of the following R code using matrix multiplication (%*%)?

R Programming

A <- matrix(c(1, 2, 3, 4), nrow=2, ncol=2, byrow=TRUE)
B <- matrix(c(2, 0, 1, 3), nrow=2, ncol=2, byrow=TRUE)
C <- A %*% B
print(C)

A

41 11 ],2[
6 4 ],1[
]2,[ ]1,[

B

[,1] [,2]
[1,] 4 8
[2,] 10 12

C

[,1] [,2]
[1,] 4 6
[2,] 11 14

D

[,1] [,2]
[1,] 5 6
[2,] 10 12

💡 Hint

Remember matrix multiplication sums the products of rows of the first matrix with columns of the second.

Matrix multiplication multiplies rows of A by columns of B and sums the products. For example, element (1,1) is 1*2 + 2*1 = 4.

❓ Predict Output

intermediate

Result of multiplying a matrix by a vector

What is the output of this R code multiplying a matrix by a vector using %*%?

R Programming

M <- matrix(c(1, 2, 3, 4, 5, 6), nrow=2, ncol=3, byrow=TRUE)
v <- c(1, 0, -1)
result <- M %*% v
print(result)

A

     [,1]
[1,]    2
[2,]    2

B

     [,1]
[1,]   -2
[2,]   -2

C

     [,1]
[1,]   -2
[2,]   -4

D

     [,1]
[1,]   -2
[2,]   -3

💡 Hint

Multiply each row of the matrix by the vector elements and sum.

M (with byrow=TRUE) is: [,1] [,2] [,3] [1,] 1 2 3 [2,] 4 5 6 First row: 1*1 + 2*0 + 3*(-1) = -2 Second row: 4*1 + 5*0 + 6*(-1) = -2

❓ Predict Output

advanced

Output of matrix multiplication with incompatible dimensions

What happens when you try to multiply these two matrices with %*% in R?

R Programming

X <- matrix(1:6, nrow=2, ncol=3)
Y <- matrix(1:4, nrow=2, ncol=2)
Z <- X %*% Y

A

[,1] [,2]
[1,] 9 12
[2,] 19 26

B NULL

C [1] 9 12 19 26

D Error in X %*% Y : non-conformable arguments

💡 Hint

Check if the number of columns in the first matrix equals the number of rows in the second.

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Here, X has 3 columns but Y has 2 rows, so multiplication is not possible.

🧠 Conceptual

advanced

Understanding the dimensions of the product matrix

If matrix A is 4x3 and matrix B is 3x5, what will be the dimensions of A %*% B in R?

A 4 rows and 5 columns

B 3 rows and 3 columns

C 5 rows and 4 columns

D 4 rows and 3 columns

💡 Hint

The resulting matrix has the number of rows of the first and columns of the second.

When multiplying matrices, the result has the number of rows of the first matrix and the number of columns of the second matrix.

❓ Predict Output

expert

Output of chained matrix multiplication with transposes

What is the output of this R code involving chained matrix multiplication and transpose?

R Programming

P <- matrix(c(1, 2, 3, 4), nrow=2)
Q <- matrix(c(0, 1, 1, 0), nrow=2)
R <- t(P) %*% Q %*% P
print(R)

A

[,1] [,2]
[1,] 4 10
[2,] 10 24

B

[,1] [,2]
[1,] 5 11
[2,] 11 25

C

[,1] [,2]
[1,] 7 13
[2,] 13 29

D

[,1] [,2]
[1,] 6 12
[2,] 12 26

💡 Hint

Calculate t(P), then multiply by Q, then multiply by P, carefully following matrix multiplication rules.

P is [[1,3],[2,4]], t(P) is [[1,2],[3,4]]. t(P) %*% Q = [[1*0+2*1, 1*1+2*0], [3*0+4*1, 3*1+4*0]] = [[2,1],[4,3]]. Then [[2,1],[4,3]] %*% P = [[2*1+1*2, 2*3+1*4], [4*1+3*2, 4*3+3*4]] = [[4,10],[10,24]].