np.linalg.det() for determinant in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Calculate determinant of a 2x2 matrix

What is the output of this code that calculates the determinant of a 2x2 matrix using np.linalg.det()?

NumPy

import numpy as np
matrix = np.array([[4, 7], [2, 6]])
det = np.linalg.det(matrix)
print(round(det, 2))

A 0.0

B 26.0

C 1.0

D 10.0

💡 Hint

Recall the formula for determinant of 2x2 matrix [[a, b], [c, d]] is ad - bc.

The determinant is calculated as (4*6) - (7*2) = 24 - 14 = 10.0.

❓ data_output

intermediate

Determinant of a singular matrix

What is the value of det after running this code for a singular matrix?

NumPy

import numpy as np
matrix = np.array([[1, 2], [2, 4]])
det = np.linalg.det(matrix)
print(det)

A 0.0

B 2.0

C 1.0

D -2.0

💡 Hint

A singular matrix has determinant zero.

The second row is twice the first, so determinant is zero.

🔧 Debug

advanced

Identify the error in determinant calculation

What error does this code raise when trying to calculate the determinant?

NumPy

import numpy as np
matrix = np.array([1, 2, 3, 4])
det = np.linalg.det(matrix)
print(det)

A ValueError: could not broadcast input array from shape (4,) into shape (2,2)

B TypeError: unsupported operand type(s) for -: 'int' and 'str'

C LinAlgError: Last 2 dimensions of the array must be square

D No error, prints 0.0

💡 Hint

Check the shape of the input array for determinant calculation.

The input is a 1D array, but determinant requires a 2D square matrix.

🚀 Application

advanced

Using determinant to check matrix invertibility

Which option correctly checks if a matrix is invertible using its determinant?

NumPy

import numpy as np
matrix = np.array([[3, 1], [2, 4]])
det = np.linalg.det(matrix)
# Check invertibility here

A

if det > 0:
    print('Invertible')
else:
    print('Not invertible')

B

if det != 0:
    print('Invertible')
else:
    print('Not invertible')

C

if det == 1:
    print('Invertible')
else:
    print('Not invertible')

D

if det < 0:
    print('Invertible')
else:
    print('Not invertible')

💡 Hint

A matrix is invertible if its determinant is not zero.

Only non-zero determinant means matrix is invertible regardless of sign.

🧠 Conceptual

expert

Effect of row operations on determinant

If you multiply one row of a square matrix by 3, how does the determinant change?

A The determinant is multiplied by 3

B The determinant remains the same

C The determinant is multiplied by 9

D The determinant becomes zero

💡 Hint

Think about how scaling a row affects volume represented by determinant.

Multiplying one row by a scalar multiplies determinant by that scalar.

Practice

1. What does the function np.linalg.det() calculate for a square matrix?

easy

A. The inverse of the matrix

B. The determinant of the matrix

C. The transpose of the matrix

D. The sum of all elements in the matrix

Solution

1. Step 1: Understand the purpose of np.linalg.det()

   This function calculates the determinant, a single number that tells if the matrix can be inverted.

2. Step 2: Compare with other matrix operations

   Transpose flips rows and columns, inverse reverses matrix multiplication, sum adds elements. These are different from determinant.

3. Final Answer:

   The determinant of the matrix -> Option B

4. Quick Check:

   np.linalg.det() = determinant [OK]

Hint: Remember: det() means determinant, not inverse or transpose [OK]

Common Mistakes:

• Confusing determinant with inverse
• Thinking it returns a matrix instead of a number
• Mixing up with transpose operation

2. Which of the following is the correct syntax to calculate the determinant of a matrix mat using numpy?

easy

A. np.linalg.determinant(mat)

B. np.det.linalg(mat)

C. np.linalg.det(mat)

D. np.det(mat)

Solution

1. Step 1: Recall the correct numpy function

   The determinant function is inside the linalg module and is called det().

2. Step 2: Check the syntax

   The correct call is np.linalg.det(mat). Other options have wrong order or function names.

3. Final Answer:

   np.linalg.det(mat) -> Option C

4. Quick Check:

   Correct syntax = np.linalg.det(mat) [OK]

Hint: Use np.linalg.det() exactly, no shortcuts [OK]

Common Mistakes:

• Swapping 'det' and 'linalg' order
• Using non-existent function names
• Omitting the linalg module

3. What is the output of the following code?

import numpy as np
mat = np.array([[2, 3], [1, 4]])
print(round(np.linalg.det(mat), 2))

medium

A. 2.0

B. 10.0

C. 11.0

D. 5.0

Solution

1. Step 1: Calculate determinant manually

   For matrix [[2,3],[1,4]], determinant = (2*4) - (3*1) = 8 - 3 = 5.

2. Step 2: Confirm output with rounding

   Rounding 5 to 2 decimals remains 5.0, matching printed output.

3. Final Answer:

   5.0 -> Option D

4. Quick Check:

   Determinant = 5.0 [OK]

Hint: Determinant 2x2 = ad - bc, calculate quickly [OK]

Common Mistakes:

• Multiplying all elements instead of ad - bc
• Forgetting to subtract
• Rounding errors without rounding function

4. The code below throws an error. What is the main reason?

import numpy as np
mat = np.array([[1, 2, 3], [4, 5, 6]])
print(np.linalg.det(mat))

medium

A. Matrix is not square

B. np.linalg.det() does not exist

C. Matrix contains integers instead of floats

D. Missing import statement

Solution

1. Step 1: Check matrix shape

   The matrix shape is (2,3), which is not square (rows != columns).

2. Step 2: Understand determinant requirements

   Determinant is defined only for square matrices, so np.linalg.det() raises an error.

3. Final Answer:

   Matrix is not square -> Option A

4. Quick Check:

   Non-square matrix causes error [OK]

Hint: Determinant needs square matrix, check shape first [OK]

Common Mistakes:

• Assuming det works on any matrix shape
• Thinking data type causes error
• Ignoring error message about shape

5. You have a 3x3 matrix:

mat = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

What does a determinant of zero imply about this matrix?

hard

A. The matrix is singular and has no inverse

B. The matrix is invertible

C. The matrix is diagonal

D. The matrix is symmetric

Solution

1. Step 1: Understand determinant zero meaning

   A zero determinant means the matrix is singular, so it cannot be inverted.

2. Step 2: Check matrix properties

   Matrix is not invertible, but zero determinant does not imply diagonal or symmetric properties.

3. Final Answer:

   The matrix is singular and has no inverse -> Option A

4. Quick Check:

   Determinant zero = no inverse [OK]

Hint: Zero determinant means no inverse exists [OK]

Common Mistakes:

• Thinking zero determinant means invertible
• Confusing diagonal or symmetric with determinant
• Ignoring singular matrix definition