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.