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.