Test yourself: Determinant properties with row/column operations questions.

📚 Quick Study Guide

• 🔢 The determinant of a matrix changes sign when two rows (or columns) are interchanged.
• ➕ Adding a multiple of one row (or column) to another row (or column) does not change the determinant.
• ⚖️ If a row (or column) is multiplied by a scalar $k$, the determinant is multiplied by $k$.
• 0️⃣ If a matrix has a row (or column) of zeros, its determinant is zero.
• 🆔 The determinant of the identity matrix is 1.
• ✖️ For a matrix $A$, if two rows or columns are identical, then $det(A) = 0$.
• ✨ $det(AB) = det(A) \cdot det(B)$

Practice Quiz

1. Question 1: If $A$ is a $3 \times 3$ matrix with $det(A) = 5$, and $B$ is obtained from $A$ by interchanging the first and third rows, then $det(B)$ is:
   1. 5
   2. -5
   3. 10
   4. 0
2. Question 2: If $A$ is a $4 \times 4$ matrix with $det(A) = 2$, and $B$ is obtained from $A$ by multiplying the second row by 3, then $det(B)$ is:
   1. 2
   2. 6
   3. 8
   4. 1/2
3. Question 3: If $A$ is a $3 \times 3$ matrix with $det(A) = -3$, and $B$ is obtained from $A$ by adding 2 times the first row to the third row, then $det(B)$ is:
   1. -3
   2. -6
   3. -1
   4. 0
4. Question 4: If $A$ is a $2 \times 2$ matrix with $det(A) = 4$, and $B$ is obtained from $A$ by multiplying $A$ by 2 (i.e., $B = 2A$), then $det(B)$ is:
   1. 4
   2. 8
   3. 16
   4. 2
5. Question 5: If $A$ is a $3 \times 3$ matrix with a row of zeros, then $det(A)$ is:
   1. 1
   2. -1
   3. 0
   4. Cannot be determined
6. Question 6: If $A$ and $B$ are $2 \times 2$ matrices with $det(A) = 2$ and $det(B) = 3$, then $det(AB)$ is:
   1. 5
   2. 6
   3. 1
   4. 0
7. Question 7: If $A$ is a $3 \times 3$ matrix, and $B$ is obtained from $A$ by swapping the first and second columns and then multiplying the third row by 4, and given that $det(A) = 1$, then $det(B)$ is:
   1. 1
   2. 4
   3. -4
   4. -1