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The question asks which of the given matrices has **no inverse**. A matrix has no inverse if it is **singular**, meaning its determinant is zero.

Let's check each matrix to see which one is singular by calculating their determinants:

1. **Matrix A**:
   \[
   A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}
   \]
   Determinant of \( A \):
   \[
   \text{det}(A) = (2)(2) - (1)(1) = 4 - 1 = 3
   \]
   Since the determinant is not zero, this matrix is **invertible**.

2. **Matrix B**:
   \[
   B = \begin{pmatrix} 5 & 0 \\ 0 & 0 \end{pmatrix}
   \]
   Determinant of \( B \):
   \[
   \text{det}(B) = (5)(0) - (0)(0) = 0
   \]
   Since the determinant is zero, this matrix is **singular** (no inverse).

3. **Matrix C**:
   \[
   C = \begin{pmatrix} 4 & 2 \\ 8 & 4 \end{pmatrix}
   \]
   Determinant of \( C \):
   \[
   \text{det}(C) = (4)(4) - (2)(8) = 16 - 16 = 0
   \]
   Since the determinant is zero, this matrix is **singular** (no inverse).

4. **Matrix D**:
   \[
   D = \begin{pmatrix} 0 & 1 \\ 0 & 2 \end{pmatrix}
   \]
   Determinant of \( D \):
   \[
   \text{det}(D) = (0)(2) - (1)(0) = 0
   \]
   Since the determinant is zero, this matrix is **singular** (no inverse).

5. **Matrix E**:
   \[
   E = \begin{pmatrix} 3 & -1 \\ 2 & 1 \end{pmatrix}
   \]
   Determinant of \( E \):
   \[
   \text{det}(E) = (3)(1) - (-1)(2) = 3 + 2 = 5
   \]
   Since the determinant is not zero, this matrix is **invertible**.

6. **Matrix F**:
   \[
   F = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
   \]
   Determinant of \( F \):
   \[
   \text{det}(F) = (0)(0) - (0)(0) = 0
   \]
   Since the determinant is zero, this matrix is **singular** (no inverse).

7. **Matrix G**:
   \[
   G = \begin{pmatrix} 6 & 6 \\ 6 & 6 \end{pmatrix}
   \]
   Determinant of \( G \):
   \[
   \text{det}(G) = (6)(6) - (6)(6) = 36 - 36 = 0
   \]
   Since the determinant is zero, this matrix is **singular** (no inverse).

8. **Matrix H**:
   \[
   H = \begin{pmatrix} 7 & 5 \\ 5 & 7 \end{pmatrix}
   \]
   Determinant of \( H \):
   \[
   \text{det}(H) = (7)(7) - (5)(5) = 49 - 25 = 24
   \]
   Since the determinant is not zero, this matrix is **invertible**.

9. **Matrix I**:
   \[
   I = \begin{pmatrix} -2 & 2 \\ 2 & 2 \end{pmatrix}
   \]
   Determinant of \( I \):
   \[
   \text{det}(I) = (-2)(2) - (2)(2) = -4 - 4 = -8
   \]
   Since the determinant is not zero, this matrix is **invertible**.

10. **Matrix J**:
    \[
    J = \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}
    \]
    Determinant of \( J \):
    \[
    \text{det}(J) = (1)(6) - (3)(2) = 6 - 6 = 0
    \]
    Since the determinant is zero, this matrix is **singular** (no inverse).

### Answer:
The matrices that have **no inverse** are:
- B
- C
- D
- F
- G
- J

Would you like more details or explanations on how determinants were calculated?

Here are 5 related questions to deepen your understanding:
1. How does the determinant help in determining whether a matrix is invertible?
2. What is the geometric significance of a matrix with zero determinant?
3. Can a non-square matrix have an inverse? Why or why not?
4. How can the adjugate matrix help find the inverse of a 2x2 matrix?
5. What is the connection between matrix rank and invertibility?

**Tip:** If you're solving a matrix inversion problem, always check if the determinant is zero first. This saves time in determining whether the inverse exists.

Which of the following matrices has no inverse?

Determine which matrices from a given set have no inverse by calculating their determinants. This exercise covers key linear algebra concepts, including the determinant formula and singular matrix theorem. Suitable for students in Grades 11-12 or undergraduate levels.

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