Solving linear systems (A\b) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of this linear system solution?

Consider the following MATLAB code that solves a system of linear equations using the backslash operator. What is the value of x after running this code?

MATLAB

A = [3 1; 1 2];
b = [9; 8];
x = A\b;

A [2; 3]

B [3; 2]

C [1; 4]

D [4; 1]

💡 Hint

Remember that x = A\b solves the system A*x = b.

Solving the system: 3*x1 + 1*x2 = 9 and 1*x1 + 2*x2 = 8 gives x1 = 2 and x2 = 3.

❓ Predict Output

intermediate

What error does this code produce?

What error will MATLAB produce when running this code?

MATLAB

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

A Warning about matrix being singular or close to singular

B Index exceeds matrix dimensions

C No error, x = [1; 1.5]

D Undefined function or variable 'x'

💡 Hint

Check if matrix A is invertible.

Matrix A is singular because the second row is a multiple of the first, so MATLAB warns that the matrix is singular or close to singular when solving.

🔧 Debug

advanced

Why does this code produce a wrong solution?

The code below attempts to solve a system but produces an incorrect solution. What is the main reason?

MATLAB

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

A Vector b has wrong dimensions

B Matrix A is not square, so the system is underdetermined or overdetermined

C Backslash operator cannot be used with numeric matrices

D Matrix A is singular

💡 Hint

Check the size of matrix A and vector b.

Matrix A is 2x3 (not square), so the system is overdetermined or underdetermined. MATLAB uses least squares solution but it may not be exact.

🧠 Conceptual

advanced

What does the MATLAB expression x = A\b compute?

In MATLAB, what does the expression x = A\b compute when A is a square, invertible matrix?

A The element-wise division of A by b

B The product of matrices A and b

C The inverse of A multiplied by b, computed explicitly

D The vector x that satisfies A*x = b

💡 Hint

Think about solving linear equations.

The backslash operator solves the linear system A*x = b efficiently without explicitly computing the inverse.

❓ Predict Output

expert

What is the output of this code with a sparse matrix?

Given the code below, what is the output of x?

MATLAB

A = sparse([1 2 3], [1 2 3], [4 5 6], 3, 3);
b = [4; 10; 18];
x = A\b;

A [0; 0; 0]

B [4; 10; 18]

C [1; 2; 3]

D Error: Sparse matrix cannot be used with backslash

💡 Hint

Sparse matrices can be used with backslash just like full matrices.

The sparse matrix A is diagonal with values 4,5,6. Solving A*x = b gives x = [1; 2; 3].

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