To determine which matrix is a diagonal matrix, let's first recall what defines a diagonal matrix. A matrix is diagonal if all its off-diagonal elements (i.e., the elements that are not on the main diagonal from the upper left to the lower right) are zero.
Here are the provided matrices:
1. [tex]\[\begin{pmatrix}2 & 0 & 0 \\ 0 & -42 & 0 \\ 0 & 16 & -7.5\end{pmatrix}\][/tex]
2. [tex]\[\begin{pmatrix}0 & 3.5 & -18 \\ 1 & 0 & 9 \\ 6 & -4 & 0\end{pmatrix}\][/tex]
3. [tex]\[\begin{pmatrix}-1 & 0 & 0 \\ 0 & -22 & 0 \\ 0 & 0 & 7.5\end{pmatrix}\][/tex]
4. [tex]\[\begin{pmatrix}0 & 0 & 7.5 \\ 0 & -22 & 0 \\ 5 & 0 & 0\end{pmatrix}\][/tex]
Let's check each matrix individually to see if all the off-diagonal elements are zero:
1. [tex]\[\begin{pmatrix}2 & 0 & 0 \\ 0 & -42 & 0 \\ 0 & 16 & -7.5\end{pmatrix}\][/tex]
- The off-diagonal elements are [tex]\(0, 0, 0, 16\)[/tex].
- Since one of the off-diagonal elements ([tex]\(16\)[/tex]) is not zero, this matrix is not a diagonal matrix.
2. [tex]\[\begin{pmatrix}0 & 3.5 & -18 \\ 1 & 0 & 9 \\ 6 & -4 & 0\end{pmatrix}\][/tex]
- The off-diagonal elements are [tex]\(3.5, -18, 1, 9, 6, -4\)[/tex].
- Since several off-diagonal elements are not zero, this matrix is not a diagonal matrix.
3. [tex]\[\begin{pmatrix}-1 & 0 & 0 \\ 0 & -22 & 0 \\ 0 & 0 & 7.5\end{pmatrix}\][/tex]
- The off-diagonal elements are [tex]\(0, 0, 0, 0, 0, 0\)[/tex].
- All the off-diagonal elements are zero, hence this matrix is a diagonal matrix.
4. [tex]\[\begin{pmatrix}0 & 0 & 7.5 \\ 0 & -22 & 0 \\ 5 & 0 & 0\end{pmatrix}\][/tex]
- The off-diagonal elements are [tex]\(0, 7.5, 5, 0\)[/tex].
- Since some of the off-diagonal elements ([tex]\(7.5\)[/tex], [tex]\(5\)[/tex]) are not zero, this matrix is not a diagonal matrix.
After examining all the matrices, we conclude that only the third matrix:
[tex]\[\begin{pmatrix}-1 & 0 & 0 \\ 0 & -22 & 0 \\ 0 & 0 & 7.5\end{pmatrix}\][/tex]
is a diagonal matrix.
