Which of the following is a diagonal matrix?
In the realm of linear algebra, diagonal matrices hold a unique position due to their simplicity and distinct properties. A diagonal matrix is a square matrix in which all the off-diagonal elements are zero, and the diagonal elements are non-zero. This characteristic makes diagonal matrices particularly useful in various mathematical applications, such as solving systems of linear equations and diagonalizing matrices. In this article, we will explore the properties of diagonal matrices and identify which of the given options is a diagonal matrix.
Diagonal matrices have several key properties that differentiate them from other types of matrices. Firstly, the determinant of a diagonal matrix is the product of its diagonal elements. This property simplifies the calculation of the determinant, as we only need to multiply the non-zero elements on the main diagonal. Secondly, the inverse of a diagonal matrix is also a diagonal matrix, with the reciprocal of each diagonal element as its corresponding entry. This property is particularly useful in solving systems of linear equations and finding eigenvalues and eigenvectors.
To determine which of the given options is a diagonal matrix, let’s consider the following examples:
Option A:
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{bmatrix}
$$
Option B:
$$
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}
$$
Option C:
$$
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}
$$
Option D:
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
$$
Upon examining the given options, we can see that Option A, Option C, and Option D are diagonal matrices. Option B is not a diagonal matrix because it has non-zero off-diagonal elements. Therefore, the correct answer to the question “Which of the following is a diagonal matrix?” is:
Option A:
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{bmatrix}
$$
Option C:
$$
\begin{bmatrix}
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & 0
\end{bmatrix}
$$
Option D:
$$
\begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
$$
Understanding the properties and characteristics of diagonal matrices is essential in linear algebra. These matrices simplify calculations and provide valuable insights into various mathematical problems.
