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].