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.

Practice

1.

What does the t() function do when applied to a matrix in R?

easy

A. It sums all elements of the matrix.

B. It calculates the inverse of the matrix.

C. It multiplies the matrix by itself.

D. It flips the matrix over its diagonal (transpose).

Solution

1. Step 1: Understand the function t()

   The t() function in R is used to transpose a matrix, which means flipping it over its diagonal.

2. Step 2: Identify the effect on the matrix

   Transposing swaps rows and columns, so the element at row i, column j moves to row j, column i.

3. Final Answer:

   It flips the matrix over its diagonal (transpose). -> Option D

4. Quick Check:

   t() = transpose [OK]

Hint: Transpose flips rows and columns across the diagonal [OK]

Common Mistakes:

• Confusing transpose with inverse
• Thinking it sums matrix elements
• Assuming it multiplies matrix by itself

2.

Which of the following is the correct syntax to find the inverse of a square matrix A in R?

easy

A. inv(A)

B. inverse(A)

C. solve(A)

D. t(A)

Solution

1. Step 1: Recall the function for matrix inverse in R

   In R, the function solve() is used to compute the inverse of a square matrix.

2. Step 2: Check the options

   Only solve(A) correctly computes the inverse. t(A) transposes, others are invalid.

3. Final Answer:

   solve(A) -> Option C

4. Quick Check:

   solve() = inverse [OK]

Hint: Use solve() to invert matrices in R [OK]

Common Mistakes:

• Using t() instead of solve() for inverse
• Assuming inverse() or inv() exist in base R
• Confusing transpose with inverse

3.

What is the output of the following R code?

A <- matrix(c(1, 2, 3, 4), nrow=2, byrow=TRUE)
B <- t(A)
B

medium

A. [1, 3]
[2, 4]

B. [1, 2]
[3, 4]

C. [4, 3]
[2, 1]

D. [2, 1]
[4, 3]

Solution

1. Step 1: Create matrix A

   Matrix A is 2x2 with rows: [1, 2] and [3, 4].

2. Step 2: Transpose matrix A to get B

   Transposing swaps rows and columns, so B's first row is [1, 3], second row is [2, 4].

3. Final Answer:

   [1, 3] [2, 4] -> Option A

4. Quick Check:

   Transpose swaps rows and columns [OK]

Hint: Transpose swaps rows and columns [OK]

Common Mistakes:

• Confusing original matrix with transpose
• Mixing row-wise and column-wise order
• Reading matrix elements incorrectly

4.

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

A <- matrix(c(1, 2, 3, 4, 5, 6), nrow=2)
invA <- solve(A)
invA

medium

A. No error; code runs fine and returns inverse.

B. Error: matrix is not square; fix by making it square (e.g., 2x2).

C. Error: solve() requires a vector; fix by passing a vector.

D. Error: matrix contains non-numeric values; fix data type.

Solution

1. Step 1: Check matrix dimensions

   Matrix A has 2 rows and 3 columns, so it is not square.

2. Step 2: Understand solve() requirements

   The solve() function requires a square matrix to compute its inverse.

3. Step 3: Fix the error

   Make A square by changing number of elements or rows/columns, e.g., 2x2 matrix.

4. Final Answer:

   Error: matrix is not square; fix by making it square (e.g., 2x2). -> Option B

5. Quick Check:

   Inverse needs square matrix [OK]

Hint: Inverse requires square matrix; check dimensions first [OK]

Common Mistakes:

• Trying to invert non-square matrix
• Ignoring error messages from solve()
• Assuming solve() works on any matrix shape

5.

You have a 3x3 matrix M. You want to check if multiplying M by its inverse returns the identity matrix. Which R code correctly performs this check?

hard

A. identity_check <- M %*% solve(M); all(identity_check == diag(3))

B. identity_check <- t(M) %*% solve(M); all(identity_check == diag(3))

C. identity_check <- solve(M) %*% M; all(identity_check == matrix(1,3,3))

D. identity_check <- M %*% M; all(identity_check == diag(3))

Solution

1. Step 1: Multiply matrix by its inverse

   Multiplying M by solve(M) should give the identity matrix.

2. Step 2: Check equality with identity matrix

   Use diag(3) to create 3x3 identity matrix and compare with result using all().

3. Step 3: Analyze options

   identity_check <- M %*% solve(M); all(identity_check == diag(3)) correctly multiplies M %*% solve(M) and compares to identity. identity_check <- t(M) %*% solve(M); all(identity_check == diag(3)) uses transpose incorrectly. identity_check <- solve(M) %*% M; all(identity_check == matrix(1,3,3)) compares to a matrix of ones, which is wrong. identity_check <- M %*% M; all(identity_check == diag(3)) multiplies M by itself, not inverse.

4. Final Answer:

   identity_check <- M %*% solve(M); all(identity_check == diag(3)) -> Option A

5. Quick Check:

   Matrix times inverse = identity [OK]

Hint: Multiply matrix by solve(matrix) to get identity [OK]

Common Mistakes:

• Using transpose instead of inverse
• Comparing to matrix of ones instead of identity
• Multiplying matrix by itself instead of inverse