Matrix inverse (inv) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of matrix inverse multiplication

What is the output of this MATLAB code snippet?

MATLAB

A = [2 1; 5 3];
B = inv(A) * A;

A [1 0; 0 1]

B [0 1; 1 0]

C [2 1; 5 3]

D Error: Matrix is singular

💡 Hint

Multiplying a matrix by its inverse results in the identity matrix.

The inverse of matrix A multiplied by A returns the identity matrix of the same size.

❓ Predict Output

intermediate

Result of inverse on singular matrix

What happens when you run this MATLAB code?

MATLAB

A = [1 2; 2 4];
B = inv(A);

A [1 2; 2 4]

B Error: Matrix is singular to working precision.

C [0.5 -0.25; -0.25 0.125]

D Error: Matrix must be square.

💡 Hint

Check if the matrix has linearly dependent rows.

Matrix A is singular because its rows are linearly dependent, so inv(A) throws an error.

🔧 Debug

advanced

Identify the error in matrix inverse calculation

What error does this MATLAB code produce?

MATLAB

A = [1 2 3; 4 5 6];
B = inv(A);

A Error: Matrix must be square.

B [0.5 -0.5 0; -0.5 0.5 0]

C [1 0 0; 0 1 0; 0 0 1]

D Error: Matrix is singular.

💡 Hint

Check the shape of the matrix before inversion.

The inv function requires a square matrix; A is 2x3, so MATLAB throws an error.

❓ Predict Output

advanced

Inverse of a diagonal matrix

What is the output of this MATLAB code?

MATLAB

D = diag([4 5 6]);
E = inv(D);

A Error: Matrix is singular.

B [4 5 6; 0 0 0; 0 0 0]

C [1 0 0; 0 1 0; 0 0 1]

D [0.25 0 0; 0 0.2 0; 0 0 0.1667]

💡 Hint

The inverse of a diagonal matrix is a diagonal matrix with reciprocal elements.

The inverse of a diagonal matrix has diagonal elements that are the reciprocals of the original.

🧠 Conceptual

expert

Properties of matrix inverse and multiplication

Given a nonsingular matrix A, which statement is TRUE?

A inv(A + B) = inv(A) + inv(B)

B inv(A * B) = inv(A) * inv(B)

C inv(A * B) = inv(B) * inv(A)

D inv(A) * A = zeros(size(A))

💡 Hint

Recall the order of multiplication when inverting a product.

The inverse of a product is the product of the inverses in reverse order: inv(A*B) = inv(B)*inv(A).

Practice

1.

What does the MATLAB function inv(A) do when A is a square matrix?

easy

A. Calculates the eigenvalues of matrix A

B. Calculates the transpose of matrix A

C. Calculates the inverse of matrix A

D. Calculates the determinant of matrix A

Solution

1. Step 1: Understand the function purpose

   The function inv in MATLAB is designed to compute the inverse of a square matrix.

2. Step 2: Recall matrix inverse definition

   The inverse of a matrix A is a matrix B such that A * B = I, where I is the identity matrix.

3. Final Answer:

   Calculates the inverse of matrix A -> Option C

4. Quick Check:

   inv(A) = inverse of A [OK]

Hint: inv() finds matrix inverse, not transpose or determinant [OK]

Common Mistakes:

• Confusing inverse with transpose
• Thinking inv() calculates determinant
• Assuming inv() returns eigenvalues

2.

Which of the following is the correct syntax to compute the inverse of a matrix A in MATLAB?

easy

A. A\inv

B. inverse(A)

C. A.inv()

D. inv(A)

Solution

1. Step 1: Recall MATLAB function for inverse

   MATLAB uses the function inv() to compute the inverse of a matrix.

2. Step 2: Check syntax correctness

   The correct syntax is inv(A). Other options are not valid MATLAB syntax for matrix inverse.

3. Final Answer:

   inv(A) -> Option D

4. Quick Check:

   Use inv(A) to invert matrix A [OK]

Hint: Use inv(A) function, not inverse() or dot notation [OK]

Common Mistakes:

• Using inverse(A) instead of inv(A)
• Trying object-oriented syntax like A.inv()
• Confusing backslash operator with inv()

3.

What is the output of the following MATLAB code?

A = [1 2; 3 4]; B = inv(A); disp(round(B,2));

medium

A. [[-2.00 1.00]; [1.50 -0.50]]

B. [[4.00 -2.00]; [-3.00 1.00]]

C. [[1.00 3.00]; [2.00 4.00]]

D. Error: Matrix is singular

Solution

1. Step 1: Calculate determinant of A

   det(A) = (1*4) - (2*3) = 4 - 6 = -2 (non-zero, so inverse exists)

2. Step 2: Compute inverse of A

   Inverse formula for 2x2 matrix: (1/det) * [d -b; -c a] = (1/-2) * [4 -2; -3 1] = [[-2 1]; [1.5 -0.5]]

3. Final Answer:

   [[-2.00 1.00]; [1.50 -0.50]] -> Option A

4. Quick Check:

   inv(A) = [[-2 1]; [1.5 -0.5]] [OK]

Hint: Use formula for 2x2 inverse: swap diagonal, negate off-diagonal [OK]

Common Mistakes:

• Mixing up signs in inverse formula
• Calculating determinant incorrectly
• Confusing inverse with transpose

4.

Consider this MATLAB code snippet:

A = [1 2; 2 4]; B = inv(A); disp(B);

What is the problem with this code?

medium

A. Matrix A is not square

B. Matrix A is singular and cannot be inverted

C. Syntax error in inv function usage

D. Matrix A is not numeric

Solution

1. Step 1: Check matrix shape

   Matrix A is 2x2, so it is square, no problem here.

2. Step 2: Check if matrix is invertible

   Calculate determinant: det(A) = (1*4) - (2*2) = 4 - 4 = 0, so A is singular and cannot be inverted.

3. Final Answer:

   Matrix A is singular and cannot be inverted -> Option B

4. Quick Check:

   det(A) = 0 means no inverse [OK]

Hint: Check determinant; zero means no inverse [OK]

Common Mistakes:

• Assuming all square matrices are invertible
• Ignoring determinant calculation
• Thinking syntax error causes failure

5.

You have a system of linear equations represented by A * x = b, where A is a 3x3 matrix and b is a vector. Which MATLAB command is the best practice to find x?

hard

A. x = A \ b;

B. x = inv(A) * b;

C. x = A / b;

D. x = b * inv(A);

Solution

1. Step 1: Understand solving linear systems in MATLAB

   Using inv(A)*b works but is inefficient and less accurate.

2. Step 2: Use MATLAB backslash operator

   The recommended way is x = A \ b; which solves the system efficiently without explicitly computing inverse.

3. Final Answer:

   x = A \ b; -> Option A

4. Quick Check:

   Use backslash \ for solving Ax=b, not inv(A) [OK]

Hint: Use backslash \ operator to solve linear systems [OK]

Common Mistakes:

• Using inv(A)*b which is slower and less stable
• Using division operators incorrectly
• Multiplying b by inv(A) instead of inv(A)*b