Let's start by solving question 1 from the uploaded image.

Question 1

Given matrices: $A = \begin{bmatrix} 3 & -4 \\ 2 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} -3 & -2 \\ -1 & 5 \end{bmatrix}, \quad C = \begin{bmatrix} 5 & 4 \\ -2 & -1 \end{bmatrix}$

Find $2A - B + 3C^T$.

1. Calculate $2A$:

$2A = 2 \times \begin{bmatrix} 3 & -4 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 6 & -8 \\ 4 & 2 \end{bmatrix}$

2. Calculate $-B$:

$-B = \begin{bmatrix} 3 & 2 \\ 1 & -5 \end{bmatrix}$

3. Calculate $3C^T$ (Transpose of $C$):

$C^T = \begin{bmatrix} 5 & -2 \\ 4 & -1 \end{bmatrix}$ $3C^T = 3 \times \begin{bmatrix} 5 & -2 \\ 4 & -1 \end{bmatrix} = \begin{bmatrix} 15 & -6 \\ 12 & -3 \end{bmatrix}$

4. Combine all parts:

$2A - B + 3C^T = \begin{bmatrix} 6 & -8 \\ 4 & 2 \end{bmatrix} + \begin{bmatrix} 3 & 2 \\ 1 & -5 \end{bmatrix} + \begin{bmatrix} 15 & -6 \\ 12 & -3 \end{bmatrix}$

Adding all these matrices element-wise:

$2A - B + 3C^T = \begin{bmatrix} 6 + 3 + 15 & -8 + 2 - 6 \\ 4 + 1 + 12 & 2 - 5 - 3 \end{bmatrix}$ $= \begin{bmatrix} 24 & -12 \\ 17 & -6 \end{bmatrix}$

So, the final result is:

$2A - B + 3C^T = \begin{bmatrix} 24 & -12 \\ 17 & -6 \end{bmatrix}$

Would you like further details or have any questions on this?

Related Questions:

1. What is the inverse of matrix $A$?
2. Calculate the determinant of matrix $B$.
3. Find the eigenvalues of matrix $C$.
4. Determine if matrix $A$ is orthogonal.
5. Find the product $AB$.

Tip: Always check the dimensions of matrices before performing operations like multiplication or addition.