https://file.aiquickdraw.com/mathbot/file/17263690641934ouwipdr.jpg

Let's work through the problems step by step from the image.

### Question 3:
Find the order of \( P^T Q^T \), where:

\[
P = \begin{pmatrix} 1 & 0 & 5 \\ 2 & 2 & 3 \end{pmatrix}, \quad Q = \begin{pmatrix} 1 & 2 \\ -1 & 3 \\ 4 & 5 \end{pmatrix}
\]

The transpose of a matrix swaps rows and columns. So:
- \( P^T \) will be a \( 3 \times 2 \) matrix (from \( 2 \times 3 \) matrix),
- \( Q^T \) will be a \( 2 \times 3 \) matrix (from \( 3 \times 2 \) matrix).

To multiply two matrices \( P^T \) and \( Q^T \), the inner dimensions must match. However, the resulting matrix \( P^T Q^T \) would have the dimensions of \( 3 \times 2 \). Thus, the answer is **B. \( 3 \times 2 \)**.

### Question 4:
Given \( P = (3 \ 5 \ 7) \) and \( Q = (9 \ -8 \ 7) \), find \( P^T Q = ? \).

Here, \( P^T \) is a column vector, and \( Q \) is a row vector. The dot product is:

\[
P^T Q = (3 \cdot 9) + (5 \cdot (-8)) + (7 \cdot 7) = 27 - 40 + 49 = 36
\]

The correct answer seems to be **D. none of these** (as the given options don't match the computed result).

### Question 5:
If \( X = \begin{pmatrix} 0 & p^2 \\ 1 & -2 - 3p \end{pmatrix} \) and \( X = -X^T \), find \( p \).

Since \( X \) is skew-symmetric (i.e., \( X = -X^T \)), we equate corresponding entries from \( X \) and \( -X^T \).

From \( X = -X^T \), the element \( p^2 \) in the first row, second column, must equal \( -(1) \) from the second row, first column. Therefore, \( p^2 = -1 \), which implies no real solution, but in complex numbers, \( p = \pm i \).

Since the options are real numbers, the correct interpretation may vary. I'll need to analyze this deeper if you'd like further.

### Question 6:
Given that \( A = \begin{pmatrix} 2 & x \\ -4 & 1 \end{pmatrix} \) and \( A^2 = \begin{pmatrix} 8 & -12 \\ -16 & 5 \end{pmatrix} \), find \( x \).

We compute \( A^2 \) by multiplying \( A \) by itself:

\[
A^2 = \begin{pmatrix} 2 & x \\ -4 & 1 \end{pmatrix} \begin{pmatrix} 2 & x \\ -4 & 1 \end{pmatrix} = \begin{pmatrix} 4 - 4x & 2x + x \\ -8 + (-4) & -4x + 1 \end{pmatrix}
\]
\[
A^2 = \begin{pmatrix} 4 - 4x & 2x + x^2 \\ -12 & -4x + 1 \end{pmatrix}
\]

Equating this with \( A^2 = \begin{pmatrix} 8 & -12 \\ -16 & 5 \end{pmatrix} \), we get:

\[
4 - 4x = 8 \quad \text{(from top left entry)}
\]
Solving \( 4 - 4x = 8 \), we get \( x = -1 \). So, the answer is **A. -1**.

### Question 7:
The square of the matrix \( \begin{pmatrix} -2 & 2 \\ 2 & a \end{pmatrix} \) is \( \begin{pmatrix} b & 0 \\ 0 & b \end{pmatrix} \), find \( a \).

Let's calculate \( A^2 \):

\[
A^2 = \begin{pmatrix} -2 & 2 \\ 2 & a \end{pmatrix} \begin{pmatrix} -2 & 2 \\ 2 & a \end{pmatrix} = \begin{pmatrix} 4 + 4 & -4 + 2a \\ -4 + 2a & 4 + a^2 \end{pmatrix}
\]
\[
A^2 = \begin{pmatrix} 8 & -4 + 2a \\ -4 + 2a & 4 + a^2 \end{pmatrix}
\]

We know \( A^2 = \begin{pmatrix} b & 0 \\ 0 & b \end{pmatrix} \), so:
- From the first equation, \( 8 = b \),
- From \( -4 + 2a = 0 \), solving for \( a \), \( a = 2 \).

Thus, the correct answer is **B. 2**.

Would you like details on any other question?

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### Follow-up Questions:
1. Can you explain why the dot product calculation for Question 4 results in an unexpected answer?
2. How do skew-symmetric matrices behave under matrix transposition?
3. Why do we multiply matrices the way we do in linear algebra?
4. Could you clarify the properties of square matrices in the context of matrix multiplication?
5. Can complex numbers be used as valid solutions for matrix entries?

**Tip:** In matrix multiplication, the dimensions of the matrices must be compatible for multiplication: the number of columns in the first matrix must equal the number of rows in the second.

Find the order of matrix operations, solve matrix equations, and calculate properties like determinants and eigenvalues.

Step-by-step solutions to matrix operations, including transposition, multiplication, and determinant calculations. Learn how to handle skew-symmetric matrices and solve eigenvalue problems in linear algebra.

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