In this problem, we are tasked with finding the values of $a$, $b$, $c$, and $d$ in the matrix equation:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} + 5 \begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T = \begin{bmatrix} -9 & -7 \\ 1 & 2 \end{bmatrix}$

Step 1: Understanding the matrix operations

We start by analyzing the second term on the left side of the equation, which is the scalar multiplication and transposition. The term $\begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T$ means we need to transpose the vector:

$\begin{bmatrix} 4 \\ 3 \\ -9 \\ x \end{bmatrix}^T = \begin{bmatrix} 4 & 3 & -9 & x \end{bmatrix}$

Then we multiply this by the scalar $5$:

$5 \begin{bmatrix} 4 & 3 & -9 & x \end{bmatrix} = \begin{bmatrix} 20 & 15 & -45 & 5x \end{bmatrix}$

Step 2: Setting up the system of equations

Now we add this result to the matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$:

$\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} 20 & 15 \\ -45 & 5x \end{bmatrix} = \begin{bmatrix} -9 & -7 \\ 1 & 2 \end{bmatrix}$

This gives us the following system of equations:

1. $a + 20 = -9$
2. $b + 15 = -7$
3. $c - 45 = 1$
4. $d + 5x = 2$

Step 3: Solving for $a$, $b$, $c$, and $d$

• From equation 1: $a = -9 - 20 = -29$
• From equation 2: $b = -7 - 15 = -22$
• From equation 3: $c = 1 + 45 = 46$
• From equation 4: $d = 2 - 5x$

To determine $d$, you need the value of $x$. Since the value for $x$ is not given, you would leave $d = 2 - 5x$ for now.

Final Answer:

• $a = -29$ (already filled correctly)
• $b = -22$
• $c = 46$
• $d = 2 - 5x$

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How does matrix transposition affect the structure of a matrix?
2. What are the steps involved in scalar multiplication of a matrix?
3. How would the solution change if $x$ were known?
4. Can this matrix operation be applied in 3D transformations?
5. What is the significance of element-wise addition in matrix equations?

Tip: Always ensure to simplify scalar operations and matrix transposition before solving for individual elements in matrix equations.