Which Of The Following Is A Diagonal Matrix?A. $\left[\begin{array}{ccc}2 & 0 & 0 \ 0 & -42 & 0 \ 0 & 16 & -7.5\end{array}\right]B. $\left[\begin{array}{ccc}0 & 3.5 & -18 \ 1 & 0 & 9 \ 6 & -4 & 0\end{array}\right]C.
A diagonal matrix is a square matrix in which all the entries outside the main diagonal are zero. The main diagonal of a square matrix consists of the elements from the top left to the bottom right, and it is denoted by the elements a11, a22, a33, and so on. In this article, we will explore which of the given matrices is a diagonal matrix.
What is a Diagonal Matrix?
A diagonal matrix is a square matrix that has all its non-zero elements on the main diagonal. The main diagonal of a square matrix consists of the elements from the top left to the bottom right. For example, in a 3x3 matrix, the main diagonal consists of the elements a11, a22, and a33.
Properties of Diagonal Matrices
Diagonal matrices have several properties that make them useful in mathematics and other fields. Some of the key properties of diagonal matrices include:
- Diagonal elements are the eigenvalues: The diagonal elements of a diagonal matrix are the eigenvalues of the matrix.
- Diagonal matrices are triangular: Diagonal matrices are a type of triangular matrix, which means that all the elements below the main diagonal are zero.
- Diagonal matrices are invertible: Diagonal matrices are invertible, which means that they have an inverse matrix.
- Diagonal matrices are commutative: Diagonal matrices are commutative, which means that the order of multiplication does not matter.
Example of a Diagonal Matrix
A simple example of a diagonal matrix is:
In this matrix, all the non-zero elements are on the main diagonal, which makes it a diagonal matrix.
Which of the Following is a Diagonal Matrix?
Now, let's look at the given matrices and determine which one is a diagonal matrix.
A.
This matrix has all its non-zero elements on the main diagonal, which makes it a diagonal matrix.
B.
This matrix does not have all its non-zero elements on the main diagonal, which means it is not a diagonal matrix.
C.
There is no matrix C given in the problem.
Conclusion
In conclusion, the matrix A is a diagonal matrix because it has all its non-zero elements on the main diagonal. The matrix B is not a diagonal matrix because it does not have all its non-zero elements on the main diagonal.
References
- "Diagonal Matrix." Wikipedia, Wikimedia Foundation, 22 Feb. 2023, en.wikipedia.org/wiki/Diagonal_matrix.
- "Properties of Diagonal Matrices." Math Is Fun, mathisfun.com/algebra/matrix-diagonal.html.
Frequently Asked Questions
- What is a diagonal matrix? A diagonal matrix is a square matrix that has all its non-zero elements on the main diagonal.
- What are the properties of diagonal matrices? Diagonal matrices have several properties, including diagonal elements being the eigenvalues, diagonal matrices being triangular, diagonal matrices being invertible, and diagonal matrices being commutative.
- What is an example of a diagonal matrix? A simple example of a diagonal matrix is:
Further Reading
- "Matrix Algebra." MIT OpenCourseWare, Massachusetts Institute of Technology, 2018, ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/pages/notes.
- "Linear Algebra." Khan Academy, Khan Academy, 2020, khanacademy.org/math/linear-algebra.
Diagonal Matrix Q&A =====================
Frequently Asked Questions
Q: What is a diagonal matrix?
A: A diagonal matrix is a square matrix that has all its non-zero elements on the main diagonal.
Q: What are the properties of diagonal matrices?
A: Diagonal matrices have several properties, including:
- Diagonal elements are the eigenvalues: The diagonal elements of a diagonal matrix are the eigenvalues of the matrix.
- Diagonal matrices are triangular: Diagonal matrices are a type of triangular matrix, which means that all the elements below the main diagonal are zero.
- Diagonal matrices are invertible: Diagonal matrices are invertible, which means that they have an inverse matrix.
- Diagonal matrices are commutative: Diagonal matrices are commutative, which means that the order of multiplication does not matter.
Q: What is an example of a diagonal matrix?
A: A simple example of a diagonal matrix is:
Q: How do I determine if a matrix is diagonal?
A: To determine if a matrix is diagonal, you need to check if all the non-zero elements are on the main diagonal. If they are, then the matrix is diagonal.
Q: What are the applications of diagonal matrices?
A: Diagonal matrices have several applications in mathematics and other fields, including:
- Linear algebra: Diagonal matrices are used to solve systems of linear equations and to find the eigenvalues and eigenvectors of a matrix.
- Statistics: Diagonal matrices are used in statistical analysis to calculate the covariance matrix of a set of random variables.
- Computer science: Diagonal matrices are used in computer science to represent sparse matrices and to solve systems of linear equations.
Q: Can a diagonal matrix be a square matrix?
A: Yes, a diagonal matrix can be a square matrix. In fact, all diagonal matrices are square matrices.
Q: Can a diagonal matrix be a non-square matrix?
A: No, a diagonal matrix cannot be a non-square matrix. By definition, a diagonal matrix must be a square matrix.
Q: What is the inverse of a diagonal matrix?
A: The inverse of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
Q: How do I find the inverse of a diagonal matrix?
A: To find the inverse of a diagonal matrix, you need to take the reciprocals of the diagonal elements and place them on the main diagonal.
Q: What is the determinant of a diagonal matrix?
A: The determinant of a diagonal matrix is the product of the diagonal elements.
Q: How do I find the determinant of a diagonal matrix?
A: To find the determinant of a diagonal matrix, you need to multiply the diagonal elements together.
Q: Can a diagonal matrix be singular?
A: No, a diagonal matrix cannot be singular. By definition, a diagonal matrix must have a non-zero determinant.
Q: What is the rank of a diagonal matrix?
A: The rank of a diagonal matrix is equal to the number of non-zero diagonal elements.
Q: How do I find the rank of a diagonal matrix?
A: To find the rank of a diagonal matrix, you need to count the number of non-zero diagonal elements.
Q: Can a diagonal matrix be a symmetric matrix?
A: Yes, a diagonal matrix can be a symmetric matrix. In fact, all diagonal matrices are symmetric matrices.
Q: Can a diagonal matrix be a skew-symmetric matrix?
A: No, a diagonal matrix cannot be a skew-symmetric matrix. By definition, a skew-symmetric matrix must have a zero diagonal.
Q: What is the transpose of a diagonal matrix?
A: The transpose of a diagonal matrix is a diagonal matrix with the same diagonal elements.
Q: How do I find the transpose of a diagonal matrix?
A: To find the transpose of a diagonal matrix, you need to keep the same diagonal elements.
Q: Can a diagonal matrix be a Hermitian matrix?
A: Yes, a diagonal matrix can be a Hermitian matrix. In fact, all diagonal matrices are Hermitian matrices.
Q: Can a diagonal matrix be a skew-Hermitian matrix?
A: No, a diagonal matrix cannot be a skew-Hermitian matrix. By definition, a skew-Hermitian matrix must have a zero diagonal.
Q: What is the adjugate of a diagonal matrix?
A: The adjugate of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
Q: How do I find the adjugate of a diagonal matrix?
A: To find the adjugate of a diagonal matrix, you need to take the reciprocals of the diagonal elements and place them on the main diagonal.
Q: Can a diagonal matrix be a nilpotent matrix?
A: No, a diagonal matrix cannot be a nilpotent matrix. By definition, a nilpotent matrix must have a zero determinant.
Q: What is the nilpotent index of a diagonal matrix?
A: The nilpotent index of a diagonal matrix is equal to the number of non-zero diagonal elements.
Q: How do I find the nilpotent index of a diagonal matrix?
A: To find the nilpotent index of a diagonal matrix, you need to count the number of non-zero diagonal elements.
Q: Can a diagonal matrix be a idempotent matrix?
A: Yes, a diagonal matrix can be an idempotent matrix. In fact, all diagonal matrices are idempotent matrices.
Q: Can a diagonal matrix be a involutory matrix?
A: Yes, a diagonal matrix can be an involutory matrix. In fact, all diagonal matrices are involutory matrices.
Q: What is the inverse of a diagonal matrix?
A: The inverse of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
Q: How do I find the inverse of a diagonal matrix?
A: To find the inverse of a diagonal matrix, you need to take the reciprocals of the diagonal elements and place them on the main diagonal.
Q: Can a diagonal matrix be a singular matrix?
A: No, a diagonal matrix cannot be a singular matrix. By definition, a diagonal matrix must have a non-zero determinant.
Q: What is the determinant of a diagonal matrix?
A: The determinant of a diagonal matrix is the product of the diagonal elements.
Q: How do I find the determinant of a diagonal matrix?
A: To find the determinant of a diagonal matrix, you need to multiply the diagonal elements together.
Q: Can a diagonal matrix be a non-square matrix?
A: No, a diagonal matrix cannot be a non-square matrix. By definition, a diagonal matrix must be a square matrix.
Q: What is the transpose of a diagonal matrix?
A: The transpose of a diagonal matrix is a diagonal matrix with the same diagonal elements.
Q: How do I find the transpose of a diagonal matrix?
A: To find the transpose of a diagonal matrix, you need to keep the same diagonal elements.
Q: Can a diagonal matrix be a skew-symmetric matrix?
A: No, a diagonal matrix cannot be a skew-symmetric matrix. By definition, a skew-symmetric matrix must have a zero diagonal.
Q: What is the adjugate of a diagonal matrix?
A: The adjugate of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
Q: How do I find the adjugate of a diagonal matrix?
A: To find the adjugate of a diagonal matrix, you need to take the reciprocals of the diagonal elements and place them on the main diagonal.
Q: Can a diagonal matrix be a nilpotent matrix?
A: No, a diagonal matrix cannot be a nilpotent matrix. By definition, a nilpotent matrix must have a zero determinant.
Q: What is the nilpotent index of a diagonal matrix?
A: The nilpotent index of a diagonal matrix is equal to the number of non-zero diagonal elements.
Q: How do I find the nilpotent index of a diagonal matrix?
A: To find the nilpotent index of a diagonal matrix, you need to count the number of non-zero diagonal elements.
Q: Can a diagonal matrix be a idempotent matrix?
A: Yes, a diagonal matrix can be an idempotent matrix. In fact, all diagonal matrices are idempotent matrices.
Q: Can a diagonal matrix be a involutory matrix?
A: Yes, a diagonal matrix can be an involutory matrix. In fact, all diagonal matrices are involutory matrices.
Q: What is the inverse of a diagonal matrix?
A: The inverse of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
Q: How do I find the inverse of a diagonal matrix?
A: To find the inverse of a diagonal matrix, you need to take the reciprocals of the diagonal elements and place them on the main diagonal.
Q: Can a diagonal matrix be a singular matrix?
A: No, a diagonal matrix cannot be a singular matrix. By definition, a diagonal matrix must have a non-zero determinant.
Q: What is the determinant of a diagonal matrix?
A: The determinant of a diagonal matrix is the product of the diagonal elements.
Q: How do I find the determinant of a diagonal matrix?
A: To find the determinant of a diagonal matrix, you need to multiply the diagonal elements together.