Transpose and inverse in R Programming - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of transpose on a matrix

What is the output of the following R code that transposes a matrix?

R Programming

mat <- matrix(1:6, nrow=2, ncol=3)
transposed <- t(mat)
print(transposed)

A

[,1] [,2]
[1,]    1    4
[2,]    2    5
[3,]    3    6

B

[,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6

C

[1,] 1 2 3
[2,] 4 5 6

D

[,1] [,2]
[1,]    4    1
[2,]    5    2
[3,]    6    3

💡 Hint

Remember that transpose flips rows and columns.

The original matrix has 2 rows and 3 columns. Transpose swaps rows and columns, so the result has 3 rows and 2 columns with elements rearranged accordingly.

❓ Predict Output

intermediate

Output of inverse of a matrix

What is the output of this R code that calculates the inverse of a matrix?

R Programming

mat <- matrix(c(4,7,2,6), nrow=2)
inv_mat <- solve(mat)
print(round(inv_mat, 2))

A Error in solve(mat) : system is computationally singular: reciprocal condition number = 0

B

     [,1] [,2]
[1,]  6.00  7.00
[2,]  2.00  4.00

C

     [,1] [,2]
[1,]  0.60 -0.20
[2,] -0.70  0.40

D

     [,1] [,2]
[1,]  0.40 -0.20
[2,] -0.70  0.60

💡 Hint

The inverse of a 2x2 matrix [[a,b],[c,d]] is (1/(ad-bc))*[[d,-b],[-c,a]].

The determinant is (4*6 - 7*2) = 24 - 14 = 10. The inverse is (1/10)*[[6,-2],[-7,4]] which is [[0.6,-0.2],[-0.7,0.4]].

🔧 Debug

advanced

Identify the error in matrix inversion code

What error will this R code produce when trying to invert a matrix?

R Programming

mat <- matrix(c(1,2,2,4), nrow=2)
inv_mat <- solve(mat)
print(inv_mat)

A Error in solve(mat) : system is computationally singular: reciprocal condition number = 0

B Error in solve(mat) : 'a' must be a square matrix

C Error in solve(mat) : non-numeric argument to mathematical function

D No error, prints inverse matrix

💡 Hint

Check if the matrix is invertible by its determinant.

The matrix has determinant zero (1*4 - 2*2 = 0), so it is singular and cannot be inverted. solve() throws this error.

🧠 Conceptual

advanced

Effect of transpose on matrix multiplication

Given two matrices A and B, which of the following is true about the transpose of their product (A %*% B)?

A t(A %*% B) equals A %*% t(B)

B t(A %*% B) equals t(A) %*% t(B)

C t(A %*% B) equals B %*% A

D t(A %*% B) equals t(B) %*% t(A)

💡 Hint

Transpose reverses the order of multiplication.

The transpose of a product of matrices equals the product of their transposes in reverse order: t(A %*% B) = t(B) %*% t(A).

❓ Predict Output

expert

Output of combined transpose and inverse operations

What is the output of this R code that combines transpose and inverse operations?

R Programming

mat <- matrix(c(2,1,1,2), nrow=2)
result <- solve(t(mat))
print(round(result, 2))

A Error in solve(t(mat)) : system is computationally singular: reciprocal condition number = 0

B

     [,1] [,2]
[1,]  0.67 -0.33
[2,] -0.33  0.67

C

     [,1] [,2]
[1,]  2.00  1.00
[2,]  1.00  2.00

D

     [,1] [,2]
[1,]  0.40 -0.20
[2,] -0.70  0.60

💡 Hint

Transpose of a symmetric matrix is the same matrix.

Since mat is symmetric, t(mat) equals mat. So solve(t(mat)) equals solve(mat), which is [[0.67, -0.33], [-0.33, 0.67]] after rounding.