Matrix multiplication (%*%) in R Programming

Introduction

Matrix multiplication helps combine two sets of numbers arranged in rows and columns to get a new set of numbers. It is useful for solving problems involving grids or tables of numbers.

When you want to combine two tables of numbers where the number of columns in the first matches the number of rows in the second.

When calculating transformations in graphics or geometry, like rotating or scaling shapes.

When working with data in statistics or machine learning that involves multiplying features by weights.

When solving systems of linear equations using matrices.

When combining multiple steps of calculations that can be represented as matrices.

Syntax

R Programming

result <- matrix1 %*% matrix2

The number of columns in matrix1 must be the same as the number of rows in matrix2.

The result will be a new matrix with rows from matrix1 and columns from matrix2.

Examples

Multiply two 2x2 matrices A and B to get matrix C.

R Programming

A <- matrix(c(1, 2, 3, 4), nrow=2, byrow=TRUE)
B <- matrix(c(5, 6, 7, 8), nrow=2, byrow=TRUE)
C <- A %*% B

Multiply a 2x3 matrix X by a 3x1 matrix Y to get a 2x1 matrix Z.

R Programming

X <- matrix(1:6, nrow=2)
Y <- matrix(7:9, nrow=3)
Z <- X %*% Y

Sample Program

This program creates two 2x2 matrices A and B, multiplies them using %*%, and prints the result matrix C.

R Programming

A <- matrix(c(1, 2, 3, 4), nrow=2, byrow=TRUE)
B <- matrix(c(5, 6, 7, 8), nrow=2, byrow=TRUE)
C <- A %*% B
print(C)

Important Notes

Matrix multiplication is not the same as element-wise multiplication (which uses * in R).

If the dimensions do not match, R will give an error.

Summary

Use %*% to multiply matrices when the inner dimensions match.

The result matrix has the outer dimensions of the two matrices.

This operation is common in math, data science, and graphics.

Practice

1. What does the operator %*% do in R?

easy

A. It multiplies two matrices using matrix multiplication rules.

B. It adds two matrices element-wise.

C. It multiplies two numbers element-wise.

D. It transposes a matrix.

Solution

1. Step 1: Understand the operator %*%

   The operator %*% in R is specifically used for matrix multiplication, not element-wise operations or addition.

2. Step 2: Differentiate from other operations

   Element-wise multiplication uses * and addition uses +. Transpose uses t(). So %*% is for matrix multiplication.

3. Final Answer:

   It multiplies two matrices using matrix multiplication rules. -> Option A

4. Quick Check:

   %*% = matrix multiplication [OK]

Hint: Remember %*% means matrix multiplication, not element-wise [OK]

Common Mistakes:

• Confusing %*% with element-wise multiplication (*)
• Thinking %*% adds matrices
• Using %*% to transpose matrices

2. Which of the following is the correct syntax to multiply two matrices A and B in R?

easy

A. A * B

B. A %/% B

C. A + B

D. A %*% B

Solution

1. Step 1: Identify matrix multiplication syntax

   In R, matrix multiplication uses the operator %*% between two matrices.

2. Step 2: Check other operators

   * is element-wise multiplication, + is addition, and %/% is integer division, so they are incorrect for matrix multiplication.

3. Final Answer:

   A %*% B -> Option D

4. Quick Check:

   Matrix multiply = A %*% B [OK]

Hint: Use %*% for matrix multiply, * is element-wise [OK]

Common Mistakes:

• Using * instead of %*% for matrix multiplication
• Using + or %/% operators mistakenly
• Forgetting spaces around %*%

3. What is the result of the following R code?

A <- matrix(c(1, 2, 3, 4), nrow=2, ncol=2)
B <- matrix(c(5, 6, 7, 8), nrow=2, ncol=2)
C <- A %*% B
print(C)

medium

A. [[6, 8], [10, 12]]

B. [[5, 12], [21, 32]]

C. [[23, 31], [34, 46]]

D. [[12, 14], [16, 18]]

Solution

1. Step 1: Construct matrices A and B (column-major order)

   A = [[1, 3], [2, 4]]
   B = [[5, 7], [6, 8]]

2. Step 2: Compute matrix multiplication

   C[1,1] = 1*5 + 3*6 = 5 + 18 = 23
   C[1,2] = 1*7 + 3*8 = 7 + 24 = 31
   C[2,1] = 2*5 + 4*6 = 10 + 24 = 34
   C[2,2] = 2*7 + 4*8 = 14 + 32 = 46

3. Final Answer:

   [[23, 31], [34, 46]] -> Option C

4. Quick Check:

   Matrix multiply result = [[23,31],[34,46]] [OK]

Hint: Remember R fills matrices by column, not row [OK]

Common Mistakes:

• Assuming row-wise filling of matrices
• Mixing element-wise and matrix multiplication
• Incorrect multiplication order

4. Consider the code below. What is the error and how to fix it?

A <- matrix(1:6, nrow=2, ncol=3)
B <- matrix(1:4, nrow=2, ncol=2)
C <- A %*% B

medium

A. Error: Syntax error; fix by adding commas in matrix() calls.

B. Error: Non-conformable arguments; fix by making A's columns equal B's rows.

C. No error; code runs fine and returns a 2x2 matrix.

D. Error: Cannot multiply matrices with different number of rows; fix by transposing B.

Solution

1. Step 1: Check matrix dimensions

   A is 2x3, B is 2x2. For matrix multiplication, A's columns (3) must equal B's rows (2). They don't match.

2. Step 2: Identify error and fix

   R will give 'non-conformable arguments' error. To fix, B must have 3 rows or A must have 2 columns. For example, B <- matrix(1:6, nrow=3, ncol=2).

3. Final Answer:

   Error: Non-conformable arguments; fix by making A's columns equal B's rows. -> Option B

4. Quick Check:

   Matrix dims must match inner dimensions [OK]

Hint: Check if A's columns = B's rows before multiplying [OK]

Common Mistakes:

• Ignoring dimension mismatch
• Trying to multiply incompatible matrices
• Assuming R auto-adjusts matrix sizes

5. You have two matrices:

M1 <- matrix(c(2, 0, 1, 3, 4, 5), nrow=2, ncol=3)
M2 <- matrix(c(1, 2, 3, 4, 5, 6), nrow=3, ncol=2)

What is the resulting matrix when you compute M1 %*% M2?

hard

A. [[16, 37], [21, 45]]

B. [[8, 12], [39, 54]]

C. [[5, 10], [29, 40]]

D. [[2, 4], [3, 6]]

Solution

1. Step 1: Construct matrices M1 and M2 (column-major order)

   M1 = [[2, 1, 4], [0, 3, 5]]
   M2 = [[1, 4], [2, 5], [3, 6]]

2. Step 2: Compute each element of the product

   C[1,1] = 2*1 + 1*2 + 4*3 = 2 + 2 + 12 = 16
   C[1,2] = 2*4 + 1*5 + 4*6 = 8 + 5 + 24 = 37
   C[2,1] = 0*1 + 3*2 + 5*3 = 0 + 6 + 15 = 21
   C[2,2] = 0*4 + 3*5 + 5*6 = 0 + 15 + 30 = 45

3. Final Answer:

   [[16, 37], [21, 45]] -> Option A

4. Quick Check:

   Matrix multiply result = [[16,37],[21,45]] [OK]

Hint: Remember R fills matrices by column, check carefully [OK]

Common Mistakes:

• Assuming row-wise filling instead of column-wise
• Mixing element-wise and matrix multiplication
• Misaligning matrix dimensions