Solving linear systems (A\b) in MATLAB - Step-by-Step Execution

Concept Flow - Solving linear systems (A\b)

Start with matrix A and vector b

↓

Check if A is square and invertible

↓

Error or alternative method

↓

Compute x = A\b

↓

Return solution vector x

↓

End

This flow shows how MATLAB solves Ax = b by checking matrix A and then computing x using the backslash operator.

Execution Sample

MATLAB

A = [2 1; 1 3];
b = [10; 15];
x = A\b;

Solves the system of equations 2x + y = 10 and x + 3y = 15 for vector x.

Execution Table

Step	Action	Evaluation	Result
1	Define matrix A	A = [2 1; 1 3]	[2 1; 1 3]
2	Define vector b	b = [10; 15]	[10; 15]
3	Check if A is square	size(A) = 2x2	True
4	Check if A is invertible	det(A) = 5	Non-zero, invertible
5	Compute x = A\b	x = A\b (internally)	[3; 4]
6	Return solution x	x	[3; 4]

💡 Computation stops after solution vector x is found.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 5	Final
A	undefined	[2 1; 1 3]	[2 1; 1 3]	[2 1; 1 3]	[2 1; 1 3]
b	undefined	undefined	[10; 15]	[10; 15]	[10; 15]
x	undefined	undefined	undefined	[3; 4]	[3; 4]

Key Moments - 2 Insights

Why do we use A\b instead of inv(A)*b?

Using A\b is more efficient and numerically stable than computing inv(A)*b explicitly, as shown in step 5 of the execution_table.

What happens if matrix A is not square or invertible?

MATLAB will either give an error or use a least-squares solution; step 3 and 4 in the execution_table check these conditions before solving.

Visual Quiz - 3 Questions

Look at the execution_table, what is the value of x after step 5?

A [2; 1]

B [3; 4]

C [8; 13]

D undefined

💡 Hint

Check the 'Result' column in row with Step 5 in execution_table.

At which step does MATLAB confirm that matrix A is invertible?

A Step 4

B Step 3

C Step 5

D Step 2

💡 Hint

Look at the 'Action' and 'Evaluation' columns in execution_table for determinant check.

If matrix A was not square, what would happen according to the concept_flow?

A MATLAB would compute x normally

B MATLAB would ignore b

C MATLAB would return an error or use an alternative method

D MATLAB would invert b

💡 Hint

Refer to the 'No' branch after 'Check if A is square and invertible' in concept_flow.

Concept Snapshot

Solving linear systems Ax = b in MATLAB:
- Use x = A\b to solve efficiently
- A must be square and invertible for exact solution
- MATLAB checks matrix properties before solving
- Avoid using inv(A)*b for better stability
- Returns solution vector x

Full Transcript

This visual execution shows how MATLAB solves linear systems using the backslash operator. First, matrix A and vector b are defined. MATLAB checks if A is square and invertible by verifying its size and determinant. If conditions are met, MATLAB computes the solution vector x using A\b, which is more efficient and stable than calculating the inverse explicitly. The variable tracker shows how A, b, and x change during execution. Key moments clarify why A\b is preferred and what happens if A is not suitable. The quiz tests understanding of solution values, matrix checks, and error handling. This helps beginners see step-by-step how MATLAB solves Ax = b.

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