Solving linear systems (A\b) in MATLAB - Mini Project: Build & Apply

Solving linear systems (A\b)

📖 Scenario: You work in a small engineering firm. You need to solve a system of linear equations to find unknown values like forces or currents.

🎯 Goal: Build a MATLAB program that solves a system of linear equations using the A\b operator.

📋 What You'll Learn

Create a matrix A with exact values

Create a vector b with exact values

Use the \ operator to solve the system

Print the solution vector x

💡 Why This Matters

🌍 Real World

Solving linear systems is common in engineering, physics, and data science to find unknown values from equations.

💼 Career

Many technical jobs require solving linear equations efficiently, and MATLAB's <code>\</code> operator is a standard tool.

1

Create matrix A

Create a 3x3 matrix called A with these exact rows:
[3, -1, 2], [2, 4, 0], [-1, 2, 5]

MATLAB

% Your code here to create matrix A

Hint

Use square brackets and semicolons to separate rows in MATLAB.

2

Create vector b

Create a column vector called b with these exact values:
5, 6, 7

MATLAB

A = [3, -1, 2; 2, 4, 0; -1, 2, 5];
% Your code here to create vector b

Hint

Use semicolons to create a column vector in MATLAB.

3

Solve the system

Create a variable called x and assign it the solution of the system using A\b

MATLAB

A = [3, -1, 2; 2, 4, 0; -1, 2, 5];
b = [5; 6; 7];
% Your code here to solve for x

Hint

Use the backslash operator \ to solve linear systems in MATLAB.

4

Display the solution

Write a disp statement to print the vector x

MATLAB

A = [3, -1, 2; 2, 4, 0; -1, 2, 5];
b = [5; 6; 7];
x = A\b;
% Your code here to display x

Hint

Use disp(x) to show the solution vector.

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