To determine the order of the matrix, we need to identify the number of rows and the number of columns in the matrix. The order of a matrix is given by the format [tex]\( m \times n \)[/tex], where [tex]\( m \)[/tex] is the number of rows and [tex]\( n \)[/tex] is the number of columns.
Let's examine the given matrix:
[tex]\[
\left[\begin{array}{ccc}
2 & -1 & 0 \\
4 & 1 & -2 \\
3 & -5 & 1 \\
3 & 6 & 1 \\
3 & 1 & 0
\end{array}\right]
\][/tex]
First, count the number of rows. A row is a horizontal line of elements in the matrix:
- The first row is [tex]\([2, -1, 0]\)[/tex]
- The second row is [tex]\([4, 1, -2]\)[/tex]
- The third row is [tex]\([3, -5, 1]\)[/tex]
- The fourth row is [tex]\([3, 6, 1]\)[/tex]
- The fifth row is [tex]\([3, 1, 0]\)[/tex]
Thus, the matrix has 5 rows.
Next, count the number of columns. A column is a vertical line of elements in the matrix:
- The first column contains the elements [tex]\(2, 4, 3, 3, 3\)[/tex]
- The second column contains the elements [tex]\(-1, 1, -5, 6, 1\)[/tex]
- The third column contains the elements [tex]\(0, -2, 1, 1, 0\)[/tex]
Thus, the matrix has 3 columns.
Therefore, the order of this matrix is [tex]\( 5 \times 3 \)[/tex].
The correct answer is:
C. [tex]\( 5 \times 3 \)[/tex]
