To rotate a matrix 90 degrees counterclockwise, we follow these steps:
1. Identify Original Matrix:
Let's start with the original matrix given:
[tex]\[
\begin{array}{cccc}
0 & 0 & -3 & 5 \\
0 & 0 & 0 & 2 \\
\end{array}
\][/tex]
2. Determine Matrix Dimensions:
The original matrix is a 2x4 matrix (2 rows and 4 columns).
3. Prepare for Rotation:
When we rotate a matrix 90 degrees counterclockwise, we'll convert each column of the original matrix into a row in the resulting matrix. The new matrix will be 4x2 (4 rows and 2 columns).
4. Fill the New Matrix:
- The first row of the new matrix will consist of the last column of the original matrix.
- The second row of the new matrix will consist of the third column of the original matrix.
- The third row of the new matrix will consist of the second column of the original matrix.
- The fourth row of the new matrix will consist of the first column of the original matrix.
Therefore, the process is as follows:
[tex]\[
\begin{array}{cccc}
0 & 0 & -3 & 5 \\
0 & 0 & 0 & 2 \\
\end{array}
\][/tex]
becomes:
[tex]\[
\begin{array}{cc}
5 & 2 \\
-3 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\][/tex]
So, after rotation, the 90-degree counterclockwise rotated matrix is:
[tex]\[
\begin{array}{cc}
5 & 2 \\
-3 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\][/tex]
This is the step-by-step process, resulting in the following matrix:
[tex]\[
\left[
\begin{array}{cc}
5 & 2 \\
-3 & 0 \\
0 & 0 \\
0 & 0 \\
\end{array}
\right]
\][/tex]
