Solved Examples of Block Matrix Determinant Calculations

📚 Quick Study Guide

🔢 Block Matrix: A matrix partitioned into submatrices called blocks. ➗ Determinant of a Block Diagonal Matrix: If $A = \begin{pmatrix} A_{11} & 0 \\ 0 & A_{22} \end{pmatrix}$, then $\det(A) = \det(A_{11}) \cdot \det(A_{22})$. ➕ Determinant of a Block Triangular Matrix: If $A = \begin{pmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{pmatrix}$ or $A = \begin{pmatrix} A_{11} & 0 \\ A_{21} & A_{22} \end{pmatrix}$, then $\det(A) = \det(A_{11}) \cdot \det(A_{22})$. 💡 Important Note: These rules apply only when the off-diagonal blocks are zero matrices or when dealing with triangular configurations.

🧪 Practice Quiz

1. Question 1: Given the block matrix $A = \begin{pmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 5 & 6 \\ 0 & 0 & 7 & 8 \end{pmatrix}$, what is $\det(A)$?
   1. -2
   2. 2
   3. -4
   4. 4
2. Question 2: Compute the determinant of the block matrix $B = \begin{pmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 1 & 3 \end{pmatrix}$.
   1. 8
   2. 10
   3. 12
   4. 14
3. Question 3: Let $C = \begin{pmatrix} 1 & 0 & 2 & 3 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$. Find $\det(C)$.
   1. 0
   2. 1
   3. 2
   4. -1
4. Question 4: Determine the determinant of $D = \begin{pmatrix} 2 & 0 & 1 & 1 \\ 0 & 2 & 1 & 1 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}$.
   1. 18
   2. 36
   3. 9
   4. 6
5. Question 5: Calculate the determinant of $E = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 2 \end{pmatrix}$.
   1. -6
   2. -3
   3. 3
   4. 6
6. Question 6: Find the determinant of the following matrix: $F = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$.
   1. 1
   2. -1
   3. 0
   4. 2
7. Question 7: Consider the block matrix $G = \begin{pmatrix} 3 & 1 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 4 & 2 \\ 0 & 0 & 2 & 4 \end{pmatrix}$. What is $\det(G)$?
   1. 72
   2. 144
   3. 36
   4. 18