https://file.aiquickdraw.com/mathbot/file/17273541784827gnkp2ks.jpg

Let's go through each of the questions:

### 11. Matrix Properties
**Question:** Let \( A \) and \( B \) be square matrices such that \( AB = 0 \). Show that if \( A \) is invertible, then \( B = C \).

- Since \( A \) is invertible, multiply both sides of \( AB = 0 \) by \( A^{-1} \):
  \[
  A^{-1}AB = A^{-1}0 \implies B = 0.
  \]
- Therefore, \( B = 0 \), which confirms \( B \) is the zero matrix.

### 12. Complex Matrices
**Given Matrices:**
\[
A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}.
\]

**a)** Finding \( A^2 \), \( A^3 \), and \( A^4 \):

1. \( A^2 = A \times A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I \).
2. \( A^3 = A^2 \times A = (-I) \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix} \).
3. \( A^4 = A^3 \times A = \begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix} \times \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I \).

**b)** Finding \( B^2 \):
\[
B^2 = B \times B = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} \times \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I.
\]

### 13. Guitar Production Increase
Given matrix:
\[
A = \begin{bmatrix} 70 & 50 & 25 \\ 35 & 100 & 70 \end{bmatrix}
\]
If production is increased by 20%, the new matrix becomes:
\[
A_{\text{new}} = 1.2 \times A = \begin{bmatrix} 1.2 \times 70 & 1.2 \times 50 & 1.2 \times 25 \\ 1.2 \times 35 & 1.2 \times 100 & 1.2 \times 70 \end{bmatrix} = \begin{bmatrix} 84 & 60 & 30 \\ 42 & 120 & 84 \end{bmatrix}.
\]

### 14. Calorie Burn Matrix
**Matrix \( A \):**
\[
A = \begin{bmatrix} 109 & 136 \\ 127 & 159 \\ 64 & 79 \end{bmatrix}
\]
- A 120-pound person and a 150-pound person bicycled for 40 minutes, jogged for 10 minutes, and walked for 60 minutes. To organize the amounts of time spent exercising in matrix \( B \):

\[
B = \begin{bmatrix} 40 & 40 \\ 10 & 10 \\ 60 & 60 \end{bmatrix}.
\]

### 15. Linear Combination of Matrices
Given matrices:
\[
X = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \quad Y = \begin{bmatrix} -1 & 0 \\ 3 & 2 \end{bmatrix}, \quad Z = \begin{bmatrix} 3 & 4 \\ -1 & 2 \end{bmatrix}, \quad W = \begin{bmatrix} 3 & 2 \\ -4 & -1 \end{bmatrix}
\]

**a)** Find scalars \( a, b, c \) such that \( W = aX + bY + cZ \).

To find \( a, b, \) and \( c \), set up the equation:
\[
W = aX + bY + cZ.
\]
Equate corresponding elements of \( W \) and solve the resulting linear system.

**b)** If \( aX + bY + cZ = 0 \), show that \( a = b = c = 0 \).

This problem can be solved by checking if the determinant of the matrix formed by \( X, Y, \) and \( Z \) (treated as vectors) is non-zero.

Let me know if you'd like a detailed explanation or solution for any of the parts above, and feel free to ask additional questions.

### Further Questions
1. How do you prove invertibility for matrices in general?
2. What are the properties of matrix multiplication that apply to the identity and zero matrices?
3. How can you determine the power of a complex matrix like \( A \) in question 12?
4. What does it mean for a matrix to be similar or share properties with another matrix?
5. How can production increases be modeled using matrix transformations?

**Tip:** When solving matrix equations, always verify the determinant and properties of the matrices involved to simplify your calculations.

The queries provided in the image including matrix operations, complex number matrices, and linear combinations of matrices.

Explore advanced matrix algebra concepts such as matrix inversion, matrix multiplication, and scalar combinations. This problem set covers topics like complex number matrices, production matrix transformations, and linear combinations of matrices. Suitable for college-level linear algebra students.

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