Δ = a(ei − fh) − b(di − fg) + c(dh − eg) for a 3x3 matrix
Δ = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) for a 3x3 matrix
|A| = |A⁻¹|⁻¹
|AB| = |A||B|
|kA| = kⁿ|A| for a nxn matrix
Cofactor expansion: |A| = a₁₁C₁₁ + a₁₂C₁₂ + ... + a₁ₙC₁ₙ
a/e = (b/f) × (c/g) for a 2x2 matrix with |A| = 0
For a 3x3 matrix, if two rows or columns are identical, then |A| = 0
If a row or column of a matrix is a linear combination of other rows or columns, then |A| = 0
If a matrix is singular, then |A| = 0

🪤 The 5 Mistakes That Cost Marks

Not checking for zero determinant before inverting a matrix
Forgetting to take the cofactor when expanding along a row or column
Not applying the properties of determinants for row or column operations
Incorrectly applying the formula for a 2x2 or 3x3 matrix
Not using the shortcut for calculating the determinant of a triangular matrix

✏️ 3 Solved PYQs

Evaluate the determinant of the matrix: | 1 2 3 | | 4 5 6 | | 7 8 9 | Using the formula for a 3x3 matrix, we get: Δ = 1(59 − 68) − 2(49 − 67) + 3(48 − 57) = 1(45 − 48) − 2 (36 − 42) + 3(32 − 35) = 1(-3) − 2(-6) + 3(-3) = -3 + 12 − 9 = 0
Find the value of x for which the determinant of the matrix is zero: | x 2 3 | | 4 5 6 | | 7 8 9 | Using the formula for a 3x3 matrix, we get: Δ = x(59 − 68) − 2(49 − 67) + 3(48 − 57) = x(45 − 48) − 2 (36 − 42) + 3(32 − 35) = x(-3) − 2(-6) + 3(-3) = -3x + 12 − 9 = -3x + 3 For |A| = 0, we have: -3x + 3 = 0 -3x = -3 x = 1
Evaluate the determinant of the matrix: | 2 0 0 | | 0 3 0 | | 0 0 4 | Using the property of determinants for a triangular matrix, we get: Δ = 234 = 24

🧠 The One Thing Most Students Get Wrong

Most students get the cofactor expansion wrong, either by not taking the cofactor or by expanding along the wrong row or column
To avoid this, always check the sign of the cofactor and make sure to expand along the correct row or column
Use the formula |A| = a₁₁C₁₁ + a₁₂C₁₂ + ... + a₁ₙC₁ₙ and make sure to calculate the cofactor correctly

👁️ Ayush's Note

To calculate the determinant of a 4x4 matrix, use the Laplace expansion
To calculate the determinant of a matrix with many rows or columns, use the LU decomposition or the Gaussian elimination method
For JEE Advanced and NEET, make sure to practice calculating the determinant of 3x3 and 4x4 matrices, as well as applying the properties of determinants for row or column operations

🔁 Last 5 Minutes Box

Check if the matrix is singular or not
Check if the matrix is a triangular matrix or not
Use the properties of determinants to simplify the calculation
Use the cofactor expansion to calculate the determinant
Check the sign of the cofactor and make sure to expand along the correct row or column

📝 Practice MCQs

1. What is the value of the determinant of the matrix:

A) 0

B) 1

C) 2

D) 3

Answer: A) The determinant of the matrix is 0, since the matrix is singular.

2. Evaluate the determinant of the matrix: | 1 2 | | 3 4 |

A) -2

B) -1

C) 0

D) 2

Answer: A) Using the formula for a 2x2 matrix, we get: Δ = 14 − 23 = 4 − 6 = -2

3. Find the value of x for which the determinant of the matrix is zero: | x 2 | | 3 4 |

A) 2

B) 3

C) 4

D) 5

Answer: B) Using the formula for a 2x2 matrix, we get: Δ = x4 − 23 = 4x − 6 For |A| = 0, we have: 4x − 6 = 0 4x = 6 x = 3/2 However, the closest option is x = 3, but the correct answer is x = 3/2, which is not among the options.

4. Evaluate the determinant of the matrix: | 1 0 0 | | 0 2 0 | | 0 0 3 |

A) 1

B) 2

C) 6

D) 12

Answer: D) Using the property of determinants for a triangular matrix, we get: Δ = 123 = 6 However, the correct answer is 6, but the option is 12, which is incorrect.

5. Find the value of the determinant of the matrix: | 2 0 0 | | 0 3 0 | | 0 0 4 |

A) 12

B) 24

C) 36

D) 48

Answer: B) Using the property of determinants for a triangular matrix, we get: Δ = 234 = 24

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Δ = a(ei − fh) − b(di − fg) + c(dh − eg) for a 3x3 matrix
Δ = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁) for a 3x3 matrix
|A| = |A⁻¹|⁻¹
|AB| = |A||B|
|kA| = kⁿ|A| for a nxn matrix
Cofactor expansion: |A| = a₁₁C₁₁ + a₁₂C₁₂ + ... + a₁ₙC₁ₙ
a/e = (b/f) × (c/g) for a 2x2 matrix with |A| = 0
For a 3x3 matrix, if two rows or columns are identical, then |A| = 0
If a row or column of a matrix is a linear combination of other rows or columns, then |A| = 0
If a matrix is singular, then |A| = 0