Sure, let's understand the structure of this matrix step by step.
We are given a 3x4 matrix with polynomial expressions inside each cell. The contents of the matrix are:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& 3x^2 & 5x & -6 \\
\hline
4x^3 & 12x^6 & 20x^4 & -24x^3 \\
\hline
2x^2 & 6x^4 & 7x^3 & -12x^2 \\
\hline
\end{array}
\][/tex]
1. First Row:
- The first element is unspecified.
- The second element is [tex]\(3x^2\)[/tex].
- The third element is [tex]\(5x\)[/tex].
- The fourth element is [tex]\(-6\)[/tex].
Putting this together, we get: [tex]\([ \text{None}, 3x^2, 5x, -6 ]\)[/tex]
2. Second Row:
- The first element is [tex]\(4x^3\)[/tex].
- The second element is [tex]\(12x^6\)[/tex].
- The third element is [tex]\(20x^4\)[/tex].
- The fourth element is [tex]\(-24x^3\)[/tex].
Putting this together, we get: [tex]\([ 4x^3, 12x^6, 20x^4, -24x^3 ]\)[/tex]
3. Third Row:
- The first element is [tex]\(2x^2\)[/tex].
- The second element is [tex]\(6x^4\)[/tex].
- The third element is [tex]\(7x^3\)[/tex].
- The fourth element is [tex]\(-12x^2\)[/tex].
Putting this together, we get: [tex]\([ 2x^2, 6x^4, 7x^3, -12x^2 ]\)[/tex]
Thus, assembling all rows, we have the complete matrix:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{None} & 3x^2 & 5x & -6 \\
\hline
4x^3 & 12x^6 & 20x^4 & -24x^3 \\
\hline
2x^2 & 6x^4 & 7x^3 & -12x^2 \\
\hline
\end{array}
\][/tex]
In summary, the given matrix with polynomial terms is:
[tex]\[
\begin{bmatrix}
\text{None} & 3x^2 & 5x & -6 \\
4x^3 & 12x^6 & 20x^4 & -24x^3 \\
2x^2 & 6x^4 & 7x^3 & -12x^2
\end{bmatrix}
\][/tex]
Let me know if you need any further clarification or assistance!
