A = [a₁₁, a₁₂, ..., a₁ₙ; a₂₁, a₂₂, ..., a₂ₙ; ...; aₙ₁, aₙ₂, ..., aₙₙ] is a matrix of order m × n
A = [aᵢⱼ]ₘₓₙ is a matrix of order m × n
A⁺ = (AᵀA)⁻¹Aᵀ is the Moore-Penrose inverse of A
|A| = a₁₁C₁ + a₁₂C₂ + ... + a₁ₙCₙ is the determinant of A
adj(A) = [Cᵢⱼ]ᵀ is the adjugate matrix of A
A⁻¹ = (1/|A|)adj(A) is the inverse of A
Aᵀ = [aⱼᵢ] is the transpose of A
AᵀA = I is the condition for orthogonal matrix A
AA⁻¹ = I is the condition for invertible matrix A
|AB| = |A||B| is the property of determinants for matrix multiplication
(AB)⁻¹ = B⁻¹A⁻¹ is the property of inverses for matrix multiplication
(Aᵀ)⁻¹ = (A⁻¹)ᵀ is the property of inverses for transpose

🪤 The 5 Mistakes That Cost Marks

Not checking the order of matrices before performing operations
Not calculating the determinant before finding the inverse
Not using the properties of determinants and inverses for matrix multiplication
Not applying the Moore-Penrose inverse for non-invertible matrices
Not using the adjugate matrix to find the inverse of a matrix

✏️ 3 Solved PYQs

Question 1: If A = [1, 2; 3, 4] and B = [5, 6; 7, 8], find |AB| and |A||B|
|AB| = | [15 + 27, 16 + 28; 35 + 47, 36 + 48] | = | [19, 22; 43, 50] | = 1950 - 2243 = 950 - 946 = 4
|A||B| = | [1, 2; 3, 4] | * | [5, 6; 7, 8] | = (14 - 23) * (58 - 67) = (-2) * (-7) = 14
Question 2: If A = [2, 1; 4, 3], find A⁻¹
|A| = 23 - 14 = 2
adj( A) = [3, -1; -4, 2]
A⁻¹ = (1/2) * [3, -1; -4, 2] = [3/2, -1/2; -2, 1]
Question 3: If A = [1, 2; 3, 4] and x = [x₁; x₂], find x if Ax = [9; 12]
[1, 2; 3, 4] * [x₁; x₂] = [9; 12]
x₁ + 2x₂ = 9
3x₁ + 4x₂ = 12
Solve the system of equations to find x₁ and x₂

🧠 The One Thing Most Students Get Wrong

Most students get the concept of matrix multiplication wrong, they think it is similar to the multiplication of numbers, but it is not, the number of columns ∈ the first matrix should be equal to the number of rows ∈ the second matrix

👁️ Ayush's Note

To find the inverse of a matrix, first check if the determinant is non-zero, if it is zero, then the matrix is not invertible
To find the determinant of a 3x3 matrix, use the formula |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
To find the adjugate matrix, find the cofactor matrix and transpose it
To find the Moore-Penrose inverse, use the formula A⁺ = (Aᵀ A) ⁻¹Aᵀ

🔁 Last 5 Minutes Box

Check the order of matrices before performing operations
Check the determinant before finding the inverse
Use the properties of determinants and inverses for matrix multiplication
Apply the Moore-Penrose inverse for non-invertible matrices
Use the adjugate matrix to find the inverse of a matrix

📝 Practice MCQs

1. Question: If A = [1, 2; 3, 4] and B = [5, 6; 7, 8], what is |AB|?

A) 2

B) 4

C) 14

D) 20

Answer: B) 4. |AB| = | [15 + 27, 16 + 28; 35 + 47, 36 + 48] | = | [19, 22; 43, 50] | = 1950 - 2243 = 950 - 946 = 4

2. Question: If A = [2, 1; 4, 3], what is A⁻¹?

A) [3/2, -1/2; -2, 1]

B) [1/2, -1/2; -2, 1]

C) [3/2, 1/2; 2, 1]

D) [1/2, 1/2; -2, 1]

Answer: A) [3/2, -1/2; -2, 1]. |A| = 23 - 14 = 2, adj(A) = [3, -1; -4, 2], A⁻¹ = (1/2) * [3, -1; -4, 2] = [3/2, -1/2; -2, 1]

3. Question: If A = [1, 2; 3, 4] and x = [x₁; x₂], what is x if Ax = [9; 12]?

A) [1; 2]

B) [2; 3]

C) [3; 4]

D) [1; 1]

Answer: B) [2; 3]. [1, 2; 3, 4] * [x₁; x₂] = [9; 12], x₁ + 2x₂ = 9, 3x₁ + 4x₂ = 12, Solve the system of equations to find x₁ and x₂

4. Question: What is the condition for a matrix to be orthogonal?

A) AᵀA = I

B) AAᵀ = I

C) AᵀA = 0

D) AAᵀ = 0

Answer: A) AᵀA = I. A matrix is orthogonal if AᵀA = I

5. Question: What is the property of inverses for matrix multiplication?

A) (A B) ⁻¹ = A⁻¹B⁻¹

B) (A B) ⁻¹ = B⁻¹A⁻¹

C) (A B) ⁻¹ = A⁻¹ + B⁻¹

D) (A B) ⁻¹ = A⁻¹ - B⁻¹

Answer: B) (AB)⁻¹ = B⁻¹A⁻¹. (AB)⁻¹ = B⁻¹A⁻¹ is the property of inverses for matrix multiplication

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

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

📚 Related Topics

Continue your revision with these related guides:

A = [a₁₁, a₁₂, ..., a₁ₙ; a₂₁, a₂₂, ..., a₂ₙ; ...; aₙ₁, aₙ₂, ..., aₙₙ] is a matrix of order m × n
A = [aᵢⱼ]ₘₓₙ is a matrix of order m × n
A⁺ = (AᵀA)⁻¹Aᵀ is the Moore-Penrose inverse of A
|A| = a₁₁C₁ + a₁₂C₂ + ... + a₁ₙCₙ is the determinant of A
adj(A) = [Cᵢⱼ]ᵀ is the adjugate matrix of A
A⁻¹ = (1/|A|)adj(A) is the inverse of A
Aᵀ = [aⱼᵢ] is the transpose of A
AᵀA = I is the condition for orthogonal matrix A
AA⁻¹ = I is the condition for invertible matrix A
|AB| = |A||B| is the property of determinants for matrix multiplication
(AB)⁻¹ = B⁻¹A⁻¹ is the property of inverses for matrix multiplication
(Aᵀ)⁻¹ = (A⁻¹)ᵀ is the property of inverses for transpose