Certainly! Let’s break down and solve the given expression step by step:
The expression to simplify is:
[tex]\[
\left(\begin{array}{ll}2k & 1\end{array}\right) \left(\begin{array}{ll} k \\ 4 \end{array}\right) - 3k(k-7)
\][/tex]
### 1. Matrix Multiplication:
First, we need to multiply the two matrices:
[tex]\[
\left(\begin{array}{ll}2k & 1\end{array}\right) \left(\begin{array}{ll} k \\ 4 \end{array}\right)
\][/tex]
The result of this multiplication is obtained by performing the dot product of corresponding elements:
[tex]\[
(2k \cdot k) + (1 \cdot 4) = 2k^2 + 4
\][/tex]
So,
[tex]\[
\left(\begin{array}{ll}2k & 1\end{array}\right) \left(\begin{array}{ll} k \\ 4 \end{array}\right) = 2k^2 + 4
\][/tex]
### 2. Simplifying the Subtraction Expression:
Next, we subtract the term [tex]\(3k(k - 7)\)[/tex]:
[tex]\[
3k(k - 7) = 3k^2 - 21k
\][/tex]
### 3. Combine Both Parts:
Now, we combine the results of our operations:
[tex]\[
2k^2 + 4 - (3k^2 - 21k)
\][/tex]
### 4. Distribute the Negative Sign:
Distribute the subtraction across the terms inside the parentheses:
[tex]\[
2k^2 + 4 - 3k^2 + 21k
\][/tex]
### 5. Combine Like Terms:
Now, combine the [tex]\(k^2\)[/tex] terms and the constant terms:
[tex]\[
(2k^2 - 3k^2) + 21k + 4 = -k^2 + 21k + 4
\][/tex]
So, the simplified result of the expression is:
[tex]\[
\boxed{-k^2 + 21k + 4}
\][/tex]
This is the final answer.
