Matrix determinant (det) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the determinant of a matrix?

The determinant is a single number that summarizes some properties of a square matrix, like whether it is invertible or not.

beginner

How do you calculate the determinant of a matrix in MATLAB?

Use the function det(A), where A is your square matrix.

intermediate

What does it mean if the determinant of a matrix is zero?

It means the matrix is singular and does not have an inverse. It also means the matrix squashes space into a lower dimension.

beginner

What is the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)?

The determinant is calculated as \( ad - bc \).

intermediate

Why is the determinant useful in real life?

It helps us understand if systems of equations have solutions, if transformations flip or shrink shapes, and in physics for volume scaling.

Which MATLAB function calculates the determinant of a matrix?

A eig()

B inv()

C rank()

D det()

The det() function returns the determinant of a square matrix.

What does a zero determinant indicate about a matrix?

A Matrix is singular

B Matrix is identity

C Matrix is diagonal

D Matrix is invertible

A zero determinant means the matrix is singular and cannot be inverted.

What is the determinant of the matrix \( \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix} \)?

A -5

B 10

C 5

D -10

Determinant = (3*1) - (4*2) = 3 - 8 = -5.

Which of these is NOT true about determinants?

A Determinant can be used to check if a matrix is invertible

B Determinant is always positive

C Determinant is a single number

D Determinant changes sign if two rows are swapped

Determinants can be positive, negative, or zero.

In MATLAB, what happens if you try to find the determinant of a non-square matrix?

A Returns 0

B Returns 1

C Throws an error

D Calculates determinant anyway

The determinant is only defined for square matrices, so MATLAB throws an error.

Explain how to calculate the determinant of a 2x2 matrix and what the result tells you about the matrix.

Think about the simple 2x2 matrix and what the determinant number means.

Describe how the MATLAB function det() is used and what kind of input it requires.

Remember the function name and input shape.

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