To determine which of the given matrices is a diagonal matrix, we need to verify that all the non-diagonal elements (those not on the main diagonal) are zero, and only the elements on the main diagonal can be non-zero.
Let's go through each matrix one by one and check if it meets the criteria for being a diagonal matrix.
Matrix 1:
[tex]\[
\begin{bmatrix}
2 & 0 & 0 \\
0 & -42 & 0 \\
0 & 16 & -7.5
\end{bmatrix}
\][/tex]
- For element (1,3) = 0 and (2,3) = 0.
- However, (3,2) = 16 which is not zero.
Since (3,2) is not zero, this matrix is not a diagonal matrix.
Matrix 2:
[tex]\[
\begin{bmatrix}
0 & 3.5 & -18 \\
1 & 0 & 9 \\
6 & -4 & 0
\end{bmatrix}
\][/tex]
- For element (1,2) = 3.5, (1,3) = -18, (2,1) = 1, (2,3) = 9, (3,1) = 6, (3,2) = -4, which are all not zero.
Since multiple non-diagonal elements are not zero, this matrix is not a diagonal matrix.
Matrix 3:
[tex]\[
\begin{bmatrix}
-1 & 0 & 0 \\
0 & -22 & 0 \\
0 & 0 & 7.5
\end{bmatrix}
\][/tex]
- For element (1,2) = 0, (1,3) = 0, (2,1) = 0, (2,3) = 0, (3,1) = 0, (3,2) = 0.
All non-diagonal elements are zero. Therefore, this matrix is a diagonal matrix.
Matrix 4:
[tex]\[
\begin{bmatrix}
0 & 0 & 7.5 \\
0 & -22 & 0
\end{bmatrix}
\][/tex]
- This is not a square matrix (it is 2x3), and diagonal matrices must be square (n x n).
Therefore, matrix 4 cannot be a diagonal matrix due to its non-square shape.
Conclusion:
Among the given matrices, only the third matrix is a diagonal matrix:
[tex]\[
\begin{bmatrix}
-1 & 0 & 0 \\
0 & -22 & 0 \\
0 & 0 & 7.5
\end{bmatrix}
\][/tex]
