Matrix inverse (inv) in MATLAB

Introduction

Matrix inverse helps you find a matrix that "undoes" the effect of the original matrix when multiplied together. It is useful to solve equations and reverse transformations.

When you want to solve a system of linear equations like Ax = b.

When you need to reverse a transformation applied by a matrix.

When calculating coefficients in linear regression or other models.

When you want to find the matrix that reverses another matrix's effect.

Syntax

MATLAB

B = inv(A)

A must be a square matrix (same number of rows and columns).

If A is not invertible (singular), MATLAB will give a warning or error.

Examples

Calculate the inverse of a 2x2 matrix A.

MATLAB

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

The inverse of the identity matrix is itself.

MATLAB

I = eye(3);
B = inv(I);

Inverse of a diagonal matrix is a diagonal matrix with reciprocal elements.

MATLAB

A = [2 0 0; 0 3 0; 0 0 4];
B = inv(A);

Sample Program

This program calculates the inverse of matrix A, then multiplies A by its inverse to show the identity matrix.

MATLAB

A = [4 7; 2 6];
B = inv(A);
disp('Matrix A:');
disp(A);
disp('Inverse of A:');
disp(B);
product = A * B;
disp('Product of A and its inverse (should be identity):');
disp(product);

Important Notes

Using inv is not always the best way to solve equations; sometimes using \ (backslash) operator is more efficient and accurate.

Only square matrices that are not singular have inverses.

Summary

The inv function finds the inverse of a square matrix.

Multiplying a matrix by its inverse gives the identity matrix.

Use inv carefully; for solving equations, prefer \.

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