Solving linear systems (A\b) in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to solve the system Ax = b.

MATLAB

x = A [1] b;

Drag options to blanks, or click blank then click option'

A\

B/

C*

D+

Attempts:

3 left

💡 Hint

Common Mistakes

Using * instead of \ to solve the system.

Using / which is for right division, not solving Ax = b.

In MATLAB, the backslash operator \ solves the system Ax = b efficiently.

2fill in blank

medium

Complete the code to solve the system and store the result in variable x.

MATLAB

x = [1] \ b;

Drag options to blanks, or click blank then click option'

Ainv(A)

Bb

Cx

DA

Attempts:

3 left

💡 Hint

Common Mistakes

Putting b before the backslash operator.

Using inv(A) which is less efficient and not recommended.

The matrix A is on the left side of the backslash operator to solve Ax = b.

3fill in blank

hard

Fix the error in the code to correctly solve the system Ax = b.

MATLAB

x = A [1] b;

Drag options to blanks, or click blank then click option'

A/

B*

C\

D+

Attempts:

3 left

💡 Hint

Common Mistakes

Using / which performs right division, not solving the system.

Using * which multiplies matrices but does not solve equations.

The correct operator to solve Ax = b is the backslash \, not division or multiplication.

4fill in blank

hard

Complete the code to create a matrix of solutions for multiple right-hand sides.

MATLAB

solutions = A [1] b(:, :);

Drag options to blanks, or click blank then click option'

A\

B:

C,

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Using * instead of \ to solve the system.

Using , instead of : to select all rows.

Use \ to solve the systems for all columns of b and : to select all rows and columns of b.

5fill in blank

hard

Fill all three blanks to solve and store solutions for each column of b in an array.

MATLAB

for [1] = 1:size(b, 2)
    solutions([2]) = A [3] b(:, i);
end

Drag options to blanks, or click blank then click option'

Ai

C\

D*

Attempts:

3 left

💡 Hint

Common Mistakes

Using * instead of \ to solve the system.

Using a different variable name than the loop index.

Use i as the loop variable and index, and \ to solve the system.

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