Matrix multiplication (%*%) in R Programming - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the operator %*% do in R?

The operator %*% performs matrix multiplication between two matrices or vectors in R. It multiplies rows of the first matrix by columns of the second matrix and sums the products.

beginner

Given matrices A (2x3) and B (3x2), what will be the dimension of A %*% B?

The result will be a 2x2 matrix because the number of rows comes from A and the number of columns comes from B.

intermediate

Why can't you multiply two matrices with %*% if the number of columns in the first doesn't equal the number of rows in the second?

Because matrix multiplication requires each element in a row of the first matrix to multiply with the corresponding element in a column of the second matrix. If sizes don't match, this pairing is impossible.

beginner

How is matrix multiplication different from element-wise multiplication in R?

Matrix multiplication (%*%) multiplies rows by columns and sums products, producing a new matrix. Element-wise multiplication (*) multiplies corresponding elements directly without summing.

beginner

What will be the output of this R code?

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

The output will be a 2x2 matrix:

     [,1] [,2]
[1,]   19   22
[2,]   43   50

Explanation: Each element is the sum of products of rows of A and columns of B.

What does the %*% operator do in R?

A Performs element-wise multiplication

B Performs matrix multiplication

C Calculates the determinant of a matrix

D Transposes a matrix

The %*% operator multiplies matrices according to matrix multiplication rules.

If A is 3x2 and B is 2x4, what is the dimension of A %*% B?

A 3x4

B 2x2

C 3x2

D 2x4

The result has rows from A and columns from B, so 3x4.

Which condition must be true to multiply matrices A and B with %*%?

A Number of columns in A equals number of rows in B

B Number of rows in A equals number of rows in B

C Number of columns in A equals number of columns in B

D Number of rows in A equals number of columns in B

Matrix multiplication requires columns of A to match rows of B.

What is the difference between * and %*% in R?

A Both do the same thing

B * is matrix multiplication; %*% is element-wise multiplication

C * is element-wise multiplication; %*% is matrix multiplication

D * transposes a matrix; %*% multiplies matrices

* multiplies elements one by one; %*% multiplies matrices properly.

What will happen if you try to multiply a 2x3 matrix by a 4x2 matrix using %*%?

A It will perform element-wise multiplication

B It will produce a 2x2 matrix

C It will produce a 4x3 matrix

D Error due to incompatible dimensions

Number of columns in first (3) does not equal number of rows in second (4), so multiplication fails.

Explain how matrix multiplication works with the %*% operator in R.

Think about how each element in the result is calculated.

Describe the difference between element-wise multiplication (*) and matrix multiplication (%*%) in R.

Consider how the two operators treat the input matrices.

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