Matrix arithmetic in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Matrix multiplication output

What is the output of this R code when multiplying two matrices?

R Programming

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

A

[,1] [,2]
[1,] 4 6
[2,] 11 12

B

[,1] [,2]
[1,] 4 9
[2,] 10 12

C

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

D

21 01 ],2[
9 4 ],1[
]2,[ ]1,[

💡 Hint

Remember matrix multiplication multiplies rows of the first by columns of the second.

Matrix multiplication multiplies each row of A by each column of B. For example, element (1,1) is 1*2 + 3*1 = 4.

❓ Predict Output

intermediate

Element-wise multiplication result

What does this R code output when performing element-wise multiplication of two matrices?

R Programming

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

A

[,1] [,2]
[1,] 2 0
[2,] 1 12

B

21 3 ],2[
0 2 ],1[
]2,[ ]1,[

C

,1] [,2]
[1,] 2 0
[2,] 3 12

D

[,1] [,2]
[1,] 2 0
[2,] 3 12

💡 Hint

Element-wise multiplication multiplies corresponding elements directly.

Each element in C is the product of corresponding elements in A and B, e.g., 1*2=2, 2*0=0, 3*1=3, 4*3=12.

❓ Predict Output

advanced

Matrix inversion and multiplication

What is the output of this R code that multiplies a matrix by its inverse?

R Programming

M <- matrix(c(4, 7, 2, 6), nrow=2)
invM <- solve(M)
result <- M %*% invM
print(round(result, 2))

A

[,1] [,2]
[1,] 1 0
[2,] 0 1

B

[,1] [,2]
[1,] 4 7
[2,] 2 6

C

[,1] [,2]
[1,] 0.5 0
[2,] 0 0.5

D

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

💡 Hint

Multiplying a matrix by its inverse should give the identity matrix.

The product of a matrix and its inverse is the identity matrix, which has 1s on the diagonal and 0s elsewhere.

❓ Predict Output

advanced

Matrix transpose and multiplication

What is the output of this R code that multiplies a matrix by its transpose?

R Programming

X <- matrix(c(1, 2, 3, 4, 5, 6), nrow=2)
Y <- X %*% t(X)
print(Y)

A

[,1] [,2]
[1,] 14 32
[2,] 32 38

B

[,1] [,2]
[1,] 14 32
[2,] 32 50

C

[,1] [,2]
[1,] 35 44
[2,] 44 56

D

[,1] [,2]
[1,] 14 20
[2,] 20 77

💡 Hint

Multiply each row of X by each row of X again because of transpose.

X is [[1,3,5],[2,4,6]]. Y = X %*% t(X). Y[1,1] = 1*1 + 3*3 + 5*5 = 35. Y[1,2] = 1*2 + 3*4 + 5*6 = 44. Symmetric matrix.

🧠 Conceptual

expert

Matrix rank and multiplication properties

Given two matrices A (3x2) and B (2x3), which statement about the rank of the product matrix C = A %*% B is always true?

A rank(C) ≤ min(rank(A), rank(B))

B rank(C) = rank(A) + rank(B)

C rank(C) ≥ max(rank(A), rank(B))

D rank(C) = rank(A) * rank(B)

💡 Hint

The rank of a product cannot exceed the ranks of the factors.

The rank of the product matrix is at most the minimum of the ranks of the two matrices multiplied.

Practice

1. In R, what does the operator %*% do when applied between two matrices?

easy

A. Performs matrix multiplication following linear algebra rules

B. Adds the two matrices element-wise

C. Subtracts the second matrix from the first element-wise

D. Multiplies the two matrices element-wise

Solution

1. Step 1: Understand matrix operators in R

   In R, + and - perform element-wise addition and subtraction respectively, while %*% is used for matrix multiplication.

2. Step 2: Identify the role of %*%

   The operator %*% multiplies matrices according to linear algebra rules, not element-wise.

3. Final Answer:

   Performs matrix multiplication following linear algebra 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 + or - does matrix multiplication
• Using * for matrix multiplication

2. Which of the following is the correct way to create a 2x2 matrix with values 1, 2, 3, 4 in R?

easy

A. matrix(1, 2, 3, 4, 2, 2)

B. matrix(1, 2, 3, 4, nrow=2, ncol=2)

C. matrix(c(1, 2, 3, 4), rows=2, cols=2)

D. matrix(c(1, 2, 3, 4), nrow=2, ncol=2)

Solution

1. Step 1: Recall matrix creation syntax in R

   The function matrix() takes a vector of values and parameters nrow and ncol to specify dimensions.

2. Step 2: Check each option

   matrix(c(1, 2, 3, 4), nrow=2, ncol=2) correctly uses c(1, 2, 3, 4) and named arguments nrow=2, ncol=2. matrix(1, 2, 3, 4, nrow=2, ncol=2) has wrong argument format. matrix(c(1, 2, 3, 4), rows=2, cols=2) uses invalid argument names. matrix(1, 2, 3, 4, 2, 2) uses positional arguments without the vector wrapper.

3. Final Answer:

   matrix(c(1, 2, 3, 4), nrow=2, ncol=2) -> Option D

4. Quick Check:

   Use c() vector and nrow, ncol named args [OK]

Hint: Always wrap values in c() and name nrow, ncol [OK]

Common Mistakes:

• Not using c() to combine values
• Using wrong argument names like rows or cols
• Passing values without vector wrapper

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

m1 <- matrix(c(1, 2, 3, 4), nrow=2, ncol=2, byrow=TRUE)
m2 <- matrix(c(5, 6, 7, 8), nrow=2, ncol=2, byrow=TRUE)
m3 <- m1 + m2
print(m3)

medium

A. [1 2; 3 4]

B. Error: non-conformable arrays

C. [6 8; 10 12]

D. [5 6; 7 8]

Solution

1. Step 1: Understand element-wise addition

   Using + between matrices adds corresponding elements: (1+5), (2+6), (3+7), (4+8).

2. Step 2: Calculate each element

   Results: 6, 8, 10, 12 arranged in 2x2 matrix.

3. Final Answer:

   [6 8; 10 12] -> Option C

4. Quick Check:

   Element-wise addition = sum of each element pair [OK]

Hint: Use + for element-wise addition, add each element pair [OK]

Common Mistakes:

• Expecting matrix multiplication instead of addition
• Confusing matrix dimensions causing errors
• Misreading matrix layout in output

4. Consider this R code snippet:

m1 <- matrix(c(1, 2, 3, 4), nrow=2)
m2 <- matrix(c(5, 6, 7), nrow=3)
m3 <- m1 %*% m2
print(m3)

What is the problem with this code?

medium

A. The matrices have incompatible dimensions for multiplication

B. The matrix m1 is not square

C. The operator %*% is invalid for matrices

D. The matrix m2 has missing values

Solution

1. Step 1: Check matrix dimensions

   m1 is 2x2, m2 is 3x1 (since 3 values and nrow=3). For multiplication, columns of m1 (2) must equal rows of m2 (3).

2. Step 2: Identify dimension mismatch

   2 != 3, so multiplication is not possible, causing an error.

3. Final Answer:

   The matrices have incompatible dimensions for multiplication -> Option A

4. Quick Check:

   Matrix multiplication needs matching inner dimensions [OK]

Hint: Check if m1 columns = m2 rows before multiplying [OK]

Common Mistakes:

• Assuming any matrices can be multiplied
• Ignoring dimension mismatch errors
• Thinking %*% works element-wise

5. You have two matrices in R:

m1 <- matrix(c(1, 0, 2, -1), nrow=2)
m2 <- matrix(c(3, 1, 4, 2), nrow=2)

You want to compute the matrix product m1 %*% m2 and then subtract the identity matrix of size 2. Which code correctly does this?

hard

A. m3 <- m1 * m2 - diag(2)

B. m3 <- m1 %*% m2 - diag(2)

C. m3 <- m1 %*% m2 - matrix(1, 2, 2)

D. m3 <- m1 %*% m2 - identity(2)

Solution

1. Step 1: Compute matrix multiplication

   Use %*% to multiply m1 and m2 correctly.

2. Step 2: Subtract identity matrix

   Use diag(2) to create a 2x2 identity matrix and subtract it from the product.

3. Final Answer:

   m3 <- m1 %*% m2 - diag(2) -> Option B

4. Quick Check:

   Use diag() for identity matrix in subtraction [OK]

Hint: Use diag(n) for identity matrix, %*% for multiplication [OK]

Common Mistakes:

• Using * instead of %*% for multiplication
• Using matrix(1,2,2) which is all ones, not identity
• Using identity() which is not an R base function