To determine the dimensions of the given matrix, we need to identify the number of rows and columns.
We start by observing the structure of the matrix:
[tex]\[
\left[\begin{array}{ccc}
4 & -2 & 2 \\
1 & 4 & 1 \\
0 & 5 & -7
\end{array}\right]
\][/tex]
Step 1: Count the number of rows:
A row is a horizontal line of elements. From the matrix above, we can see:
- The first row is [tex]\( [4, -2, 2] \)[/tex]
- The second row is [tex]\( [1, 4, 1] \)[/tex]
- The third row is [tex]\( [0, 5, -7] \)[/tex]
Thus, there are 3 rows in this matrix.
Step 2: Count the number of columns:
A column is a vertical line of elements. From the matrix above, we can count the number of elements in each row, and verify they are consistent across all rows:
- The first column has [tex]\( 4, 1, 0 \)[/tex]
- The second column has [tex]\( -2, 4, 5 \)[/tex]
- The third column has [tex]\( 2, 1, -7 \)[/tex]
Thus, there are 3 columns in this matrix.
Conclusion:
The matrix has 3 rows and 3 columns. Therefore, its dimensions are [tex]\( 3 \times 3 \)[/tex].
So, the correct answer is:
[tex]\[ 3 \times 3 \][/tex]
