Step-by-Step Guide to Verifying det(Aᵀ) = det(A) with Examples

📚 Quick Study Guide

• 🔢 The transpose of a matrix, denoted as $A^T$, is obtained by interchanging its rows and columns.
• 🧮 The determinant of a matrix, denoted as $det(A)$, is a scalar value computed from the elements of a square matrix.
• 💡 The key property is: For any square matrix $A$, $det(A^T) = det(A)$. This means the determinant of a matrix is equal to the determinant of its transpose.
• 📝 This property holds true for all square matrices, regardless of their size or the values of their elements.
• ➗ The determinant can be calculated using various methods, such as cofactor expansion or row reduction. The choice of method does not affect the final value of the determinant.

Practice Quiz

1. Question 1: Which of the following statements is always true regarding the determinant of a matrix and its transpose?
1. A) $det(A^T) = -det(A)$
2. B) $det(A^T) = det(A)$
3. C) $det(A^T) = 1/det(A)$
4. D) $det(A^T) = det(A)^2$
2. Question 2: If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, what is $det(A^T)$?
1. A) -2
2. B) 2
3. C) 10
4. D) -10
3. Question 3: Given a matrix $B$ where $det(B) = 5$, what is $det(B^T)$?
1. A) -5
2. B) 1/5
3. C) 5
4. D) 25
4. Question 4: Which operation describes obtaining $A^T$ from $A$?
1. A) Multiplying each element by -1
2. B) Swapping rows and columns
3. C) Adding the matrix to its inverse
4. D) Taking the square root of each element
5. Question 5: If $C = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $C^T$ is?
1. A) $\begin{bmatrix} c & d \\ a & b \end{bmatrix}$
2. B) $\begin{bmatrix} b & a \\ d & c \end{bmatrix}$
3. C) $\begin{bmatrix} a & c \\ b & d \end{bmatrix}$
4. D) $\begin{bmatrix} d & c \\ b & a \end{bmatrix}$
6. Question 6: The property $det(A^T) = det(A)$ holds for:
1. A) Only 2x2 matrices
2. B) Only 3x3 matrices
3. C) All square matrices
4. D) Only invertible matrices
7. Question 7: If $det(A) = 0$, what can you say about $det(A^T)$?
1. A) $det(A^T)$ is undefined
2. B) $det(A^T) = 1$
3. C) $det(A^T) = 0$
4. D) $det(A^T)$ cannot be determined