Solving linear systems (A\b) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the MATLAB expression A\b do?

It solves the linear system of equations A*x = b for the vector x. It finds x such that when multiplied by A, it equals b.

intermediate

Why is A\b preferred over directly calculating inv(A)*b?

Because A\b is more efficient and numerically stable. It uses optimized algorithms to solve the system without explicitly computing the inverse, which can be slow and less accurate.

intermediate

What happens if matrix A is singular or nearly singular when using A\b?

MATLAB will give a warning or error because the system may not have a unique solution. It tries to find a least-squares solution if possible, but the result might be unreliable.

intermediate

How can you solve a system with multiple right-hand sides using A\b?

If b is a matrix with multiple columns, A\b solves each system A*x = b(:,i) simultaneously, returning a matrix of solutions.

beginner

What type of matrix is required for A\b to work correctly?

Matrix A should be square (same number of rows and columns) and non-singular (invertible) for a unique solution. For non-square matrices, A\b finds a least-squares solution.

What does x = A\b compute in MATLAB?

A Solution to the system A*x = b

B Inverse of matrix A times b

C Determinant of A

D Transpose of A times b

A\b solves the linear system A*x = b for x.

Why is A\b preferred over inv(A)*b in MATLAB?

A It is slower but more accurate

B It computes the inverse explicitly

C It is faster and more numerically stable

D It only works for diagonal matrices

A\b uses optimized algorithms that avoid computing the inverse, making it faster and more stable.

What happens if A is singular when using A\b?

A MATLAB returns zero vector

B MATLAB computes the exact solution

C MATLAB ignores the singularity

D MATLAB returns an error or warning

Singular matrices do not have unique solutions, so MATLAB warns or errors when solving.

If b has multiple columns, what does A\b do?

A Solves each system for each column of <code>b</code>

B Only solves for the first column of <code>b</code>

C Returns an error

D Computes the inverse of <code>A</code> times <code>b</code>

A\b solves multiple systems simultaneously when b has multiple columns.

What kind of matrix A is ideal for using A\b to get a unique solution?

A Singular

B Square and invertible

C Rectangular

D Zero matrix

A square and invertible matrix ensures a unique solution for A\b.

Explain in your own words how MATLAB solves a system of linear equations using A\b.

Think about what the backslash operator means in MATLAB.

Describe what happens if the matrix A is not square when you use A\b.

Consider how MATLAB handles systems with more equations than unknowns or vice versa.

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