Matrix determinant (det) in MATLAB - Step-by-Step Execution

Concept Flow - Matrix determinant (det)

Start with matrix A

↓

Check matrix size

↓

If 1x1 matrix

Yes→Return element as determinant

No↓

For 2x2 matrix

Yes→Calculate ad - bc

No↓

For larger matrix

↓

Expand along a row or column

↓

Calculate minors and cofactors

↓

Recursive call det on minors

↓

Sum products of elements and cofactors

↓

Return determinant value

↓

End

The determinant is calculated by checking matrix size, using simple formulas for small matrices, and recursively expanding for larger ones.

Execution Sample

MATLAB

A = [1 2; 3 4];
d = det(A);
disp(d);

Calculates and displays the determinant of a 2x2 matrix A.

Execution Table

Step	Matrix	Action	Calculation	Result
1	[1 2; 3 4]	Check size	2x2 matrix	Proceed to 2x2 formula
2	[1 2; 3 4]	Calculate determinant	1*4 - 2*3	-2
3	-	Return determinant	-	-2

💡 Determinant calculated for 2x2 matrix using formula ad - bc

Variable Tracker

Variable	Start	After Step 2	Final
A	[1 2; 3 4]	[1 2; 3 4]	[1 2; 3 4]
d	undefined	-2	-2

Key Moments - 2 Insights

Why do we use the formula ad - bc for 2x2 matrices?

Because for 2x2 matrices, the determinant is defined as ad - bc, as shown in execution_table step 2.

What happens if the matrix is larger than 2x2?

The determinant is calculated by expanding along a row or column recursively, not shown in this simple example but described in concept_flow.

Visual Quiz - 3 Questions

Look at the execution_table, what is the determinant value after step 2?

A 2

B -2

C 0

D Undefined

💡 Hint

Check the 'Result' column in execution_table row 2.

At which step does the program decide to use the 2x2 determinant formula?

A Step 3

B Step 2

C Step 1

D No step

💡 Hint

Look at the 'Action' and 'Calculation' columns in execution_table row 1.

If matrix A was 1x1, what would the determinant be according to concept_flow?

A The single element of the matrix

B Zero

C One

D Not defined

💡 Hint

Refer to concept_flow where 1x1 matrix returns the element as determinant.

Concept Snapshot

Matrix determinant (det) in MATLAB:
- Use det(A) to find determinant.
- For 1x1 matrix, determinant is the element.
- For 2x2 matrix, use ad - bc.
- For larger matrices, recursive expansion is used.
- Determinant is a single number representing matrix scaling.

Full Transcript

This visual execution traces how MATLAB calculates the determinant of a matrix. Starting with matrix A, the program checks its size. For a 2x2 matrix like [1 2; 3 4], it uses the formula ad - bc, calculating 1*4 - 2*3 = -2. The determinant value -2 is returned. For larger matrices, MATLAB would expand along a row or column recursively, calculating minors and cofactors, but this example focuses on the simple 2x2 case. Variables A and d are tracked, showing d changes from undefined to -2 after calculation. Common confusions include why the formula ad - bc is used for 2x2 matrices and what happens for larger matrices. The quiz questions check understanding of these steps and values.

Practice

1.

What does the MATLAB function det(A) compute when A is a square matrix?

easy

A. The determinant of matrix A

B. The transpose of matrix A

C. The inverse of matrix A

D. The sum of all elements in matrix A

Solution

1. Step 1: Understand the function purpose

   The function det(A) in MATLAB calculates a single number called the determinant for a square matrix A.

2. Step 2: Recall determinant meaning

   The determinant tells us about matrix properties like invertibility, not transpose, inverse, or sum.

3. Final Answer:

   The determinant of matrix A -> Option A

4. Quick Check:

   det(A) = determinant [OK]

Hint: Remember det() always returns a single number for square matrices [OK]

Common Mistakes:

• Confusing determinant with transpose
• Thinking det() returns a matrix
• Mixing determinant with sum of elements

2.

Which of the following is the correct MATLAB syntax to find the determinant of matrix M?

?

easy

A. determinant(M)

B. det(M)

C. determinant = M

D. determinant(M);

Solution

1. Step 1: Identify MATLAB function for determinant

   The correct MATLAB function to compute determinant is det().

2. Step 2: Check syntax correctness

   The correct syntax is det(M). Other options use wrong function names or incorrect assignment.

3. Final Answer:

   det(M) -> Option B

4. Quick Check:

   Use det(M) to get determinant [OK]

Hint: Use det() exactly, no extra words or assignments [OK]

Common Mistakes:

• Using 'determinant' instead of 'det'
• Adding semicolon inside function call
• Assigning determinant without function

3.

What is the output of the following MATLAB code?

A = [2 3; 1 4];
d = det(A);
d

medium

A. -5

B. 11

C. 10

D. 5

Solution

1. Step 1: Calculate determinant of matrix A

   Matrix A = [2 3; 1 4]. Determinant = (2*4) - (3*1) = 8 - 3 = 5.

2. Step 2: Assign and display determinant

   Variable d stores 5, so output is 5.

3. Final Answer:

   5 -> Option D

4. Quick Check:

   det([2 3;1 4]) = 5 [OK]

Hint: Use formula ad-bc for 2x2 matrix determinant [OK]

Common Mistakes:

• Multiplying elements incorrectly
• Swapping terms in subtraction
• Confusing determinant with sum

4.

Identify the error in this MATLAB code snippet:

B = [1 2 3; 4 5 6];
detB = det(B);

medium

A. Variable detB is not defined properly

B. Syntax error in det function call

C. Matrix B is not square, so det(B) causes an error

D. No error, code runs fine

Solution

1. Step 1: Check matrix dimensions

   Matrix B is 2x3, which is not square.

2. Step 2: Understand det() requirements

   det() requires a square matrix; using it on non-square matrix causes an error.

3. Final Answer:

   Matrix B is not square, so det(B) causes an error -> Option C

4. Quick Check:

   det() needs square matrix [OK]

Hint: Only square matrices have determinants [OK]

Common Mistakes:

• Trying det() on non-square matrix
• Ignoring matrix size requirements
• Assuming det() returns zero for non-square

5.

Given matrix C = [1 2 3; 0 1 4; 5 6 0], which MATLAB code correctly checks if C is invertible using its determinant?

hard

A. if det(C) ~= 0
disp('Invertible')
else
disp('Not invertible')
end

B. if det(C) == 0
disp('Invertible')
else
disp('Not invertible')
end

C. if det(C) > 1
disp('Invertible')
else
disp('Not invertible')
end

D. if det(C) < 0
disp('Invertible')
else
disp('Not invertible')
end

Solution

1. Step 1: Recall invertibility condition

   A matrix is invertible if its determinant is not zero.

2. Step 2: Analyze code options

   if det(C) ~= 0 disp('Invertible') else disp('Not invertible') end checks if det(C) ~= 0, correctly identifying invertibility. Others check wrong conditions.

3. Final Answer:

   if det(C) ~= 0 disp('Invertible') else disp('Not invertible') end -> Option A

4. Quick Check:

   Invertible if det ≠ 0 [OK]

Hint: Invertible means determinant not zero [OK]

Common Mistakes:

• Checking det == 0 for invertibility
• Using > 1 or < 0 instead of zero check
• Confusing invertible with positive determinant