Solving linear systems (A\b) in MATLAB

Introduction

We use A\b to find the values that solve equations like Ax = b. It helps us quickly find unknowns in math problems.

When you have a set of equations and want to find the unknown values.

When you want to solve problems like finding forces, currents, or prices that fit given conditions.

When you want a fast and easy way to solve many equations at once.

When you want to check if a solution exists or find the best approximate answer.

Syntax

MATLAB

x = A \ b

A is a matrix representing coefficients of the equations.

b is a column vector representing the results of the equations.

Examples

Solves two equations with two unknowns.

MATLAB

A = [2 1; 3 4];
b = [5; 6];
x = A \ b;

Solves three equations with three unknowns.

MATLAB

A = [1 2 3; 4 5 6; 7 8 10];
b = [6; 15; 25];
x = A \ b;

Solves equations where A is singular or has infinite solutions; MATLAB gives a least squares solution.

MATLAB

A = [1 2; 2 4];
b = [3; 6];
x = A \ b;

Sample Program

This program solves the system of equations:
3x + 2y = 7
x + 4y = 10
and shows the values of x and y.

MATLAB

A = [3 2; 1 4];
b = [7; 10];
x = A \ b;
disp('Solution x:');
disp(x);

Important Notes

Use A\b instead of inv(A)*b for better speed and accuracy.

If A is not square or singular, MATLAB finds the best approximate solution.

Always check the size of A and b to match the system you want to solve.

Summary

A\b solves equations Ax = b quickly and easily.

It works for square and some non-square systems.

It is more reliable than calculating the inverse of A.

Practice

1. What does the MATLAB expression x = A\b do?

easy

A. Solves the linear system Ax = b for x

B. Calculates the inverse of matrix A

C. Multiplies matrix A by vector b

D. Transposes matrix A and multiplies by b

Solution

1. Step 1: Understand the expression A\b

   The operator \ in MATLAB is used to solve linear systems of the form Ax = b.

2. Step 2: Identify the purpose of A\b

   It finds the vector x that satisfies the equation Ax = b without explicitly calculating the inverse of A.

3. Final Answer:

   Solves the linear system Ax = b for x -> Option A

4. Quick Check:

   A\b solves Ax = b [OK]

Hint: Remember: A\b solves Ax = b directly [OK]

Common Mistakes:

• Confusing A\b with matrix multiplication
• Thinking it calculates inverse of A
• Using b\A instead of A\b

2. Which of the following is the correct syntax to solve the system Ax = b in MATLAB?

easy

A. x = A\b;

B. x = b/A;

C. x = inv(A)\b;

D. x = A*b;

Solution

1. Step 1: Recall MATLAB syntax for solving linear systems

   The correct syntax uses the backslash operator: x = A\b;

2. Step 2: Check other options for correctness

   x = b/A; uses b/A which is incorrect; x = inv(A)\b; uses inv(A)\b which solves inv(A)x = b (wrong system); x = A*b; multiplies A and b which is not solving the system.

3. Final Answer:

   x = A\b; -> Option A

4. Quick Check:

   Use backslash operator for solving Ax = b [OK]

Hint: Use backslash \ operator, not division or inverse [OK]

Common Mistakes:

• Using b/A instead of A\b
• Using inv(A)\b which solves wrong system
• Confusing multiplication with solving

3. Given A = [2 1; 3 4] and b = [5; 10], what is the result of x = A\b?

medium

A. [1; 2]

B. [5; 6]

C. [0; 0]

D. [2; 1]

Solution

1. Step 1: Write the system of equations

   2x1 + x2 = 5
   3x1 + 4x2 = 10.

2. Step 2: Solve using elimination

   Multiply first equation by 4: 8x1 + 4x2 = 20.
   Subtract second: 5x1 = 10 -> x1 = 2.
   Substitute: x2 = 5 - 2*2 = 1.

3. Step 3: Verify solution

   A*[2; 1] = [2*2 + 1*1; 3*2 + 4*1] = [5; 10] = b.

4. Final Answer:

   [2; 1] -> Option D

5. Quick Check:

   A\b = [2; 1] [OK]

Hint: Use A\b to get exact solution vector [OK]

Common Mistakes:

• Miscalculating substitution (e.g., picking [1; 2])
• Confusing with matrix multiplication A*b
• Selecting x = b

4. What is wrong with this MATLAB code snippet?

A = [1 2; 3 4];
b = [5; 6; 7];
x = A\b;

medium

A. Backslash operator cannot solve linear systems

B. Matrix A and vector b have incompatible dimensions

C. Missing semicolon after assignment

D. Matrix A is not square

Solution

1. Step 1: Check dimensions of A and b

   Matrix A is 2x2, vector b is 3x1. For Ax = b, number of rows in A must equal length of b.

2. Step 2: Identify dimension mismatch

   Since A has 2 rows but b has 3 elements, dimensions are incompatible, causing an error when solving.

3. Final Answer:

   Matrix A and vector b have incompatible dimensions -> Option B

4. Quick Check:

   Rows of A must match length of b [OK]

Hint: Check matrix and vector sizes before solving [OK]

Common Mistakes:

• Ignoring dimension mismatch errors
• Thinking backslash works for any sizes
• Assuming A must be square always

5. You have a non-square matrix A = [1 2 3; 4 5 6] and vector b = [7; 8]. What does x = A\b compute in MATLAB?

hard

A. The inverse of A times b

B. An error because A is not square

C. A least-squares solution minimizing ||Ax - b||

D. A random vector x

Solution

1. Step 1: Recognize matrix shape and operation

   A is 2x3 (more columns than rows), so system is underdetermined.

2. Step 2: Understand MATLAB's backslash behavior for non-square A

   MATLAB computes the least-squares solution x that minimizes ||Ax - b|| (minimum norm among solutions).

3. Final Answer:

   A least-squares solution minimizing ||Ax - b|| -> Option C

4. Quick Check:

   Backslash solves least-squares for non-square A [OK]

Hint: Backslash gives least-squares solution if A not square [OK]

Common Mistakes:

• Expecting error for non-square A
• Thinking it calculates inverse always
• Assuming exact solution always exists