Matrix inverse (inv) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the inv function do in MATLAB?

The inv function calculates the inverse of a square matrix, if it exists. The inverse matrix, when multiplied by the original, gives the identity matrix.

beginner

What is the identity matrix in the context of matrix inversion?

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. Multiplying any matrix by the identity matrix leaves it unchanged.

beginner

What happens if you try to invert a non-square matrix using inv in MATLAB?

MATLAB will give an error because only square matrices can have an inverse.

intermediate

How can you check if a matrix is invertible before using inv?

You can check if the matrix determinant is not zero using det(A) ~= 0. If the determinant is zero, the matrix is singular and not invertible.

intermediate

Why is it better to use matrix division operators (\ or /) instead of inv for solving linear systems?

Using matrix division operators is more efficient and numerically stable than calculating the inverse explicitly. It avoids unnecessary computations and reduces rounding errors.

What is the result of multiplying a matrix by its inverse?

A A matrix of ones

B A zero matrix

C The original matrix squared

D The identity matrix

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

Which MATLAB function is used to find the inverse of a matrix?

A inv

B det

C transpose

D eig

The inv function calculates the inverse of a matrix.

What condition must a matrix meet to have an inverse?

A It must have all positive elements

B It must be square and have a non-zero determinant

C It must be rectangular

D It must be symmetric

Only square matrices with non-zero determinant are invertible.

What error occurs if you try to invert a singular matrix in MATLAB?

A Index exceeds matrix dimensions

B Undefined function or variable

C Matrix is singular to working precision

D Syntax error

MATLAB reports the matrix is singular if it cannot be inverted.

Which is the recommended way to solve linear equations in MATLAB?

A Using matrix division operators (\ or /)

B Using inv function

C Using loops

D Using transpose function

Matrix division operators are more efficient and stable than using inv.

Explain how to find the inverse of a matrix in MATLAB and what conditions must be met for the inverse to exist.

Think about what makes a matrix invertible and how MATLAB checks it.

Describe why using matrix division operators is preferred over the inv function for solving linear systems.

Consider what happens behind the scenes when solving equations.

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