Det(A)

Definition

The determinant of a square matrix A, denoted as det(A) or |A|, is a scalar value that provides important information about the matrix, such as whether it is invertible and how it transforms the space. The determinant is a fundamental concept in linear algebra that has numerous applications in mathematics, physics, and other fields.

5 Must Know Facts For Your Next Test

1. The determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is given by the formula: $det(A) = ad - bc$.
2. If det(A) = 0, then the matrix A is singular and does not have an inverse. If det(A) ≠ 0, then A is invertible, and its inverse is given by $A^{-1} = \frac{1}{det(A)} adj(A)$, where $adj(A)$ is the adjoint of A.
3. The determinant of a matrix is a multilinear function of the rows (or columns) of the matrix, meaning that it is linear in each row (or column) and the determinant changes linearly when a row (or column) is multiplied by a scalar.
4. The determinant of a diagonal matrix is the product of its diagonal elements, and the determinant of a triangular matrix is the product of its diagonal elements.
5. The determinant of a matrix A is related to the volume of the parallelotope spanned by the column (or row) vectors of A in n-dimensional space.

Review Questions

• Explain the relationship between the determinant of a matrix and the existence of its inverse.
  • The determinant of a square matrix A, denoted as det(A), is closely related to the existence of the matrix inverse, A^-1. If det(A) ≠ 0, then the matrix A is invertible, and its inverse is given by the formula A^-1 = (1/det(A)) * adj(A), where adj(A) is the adjoint of A. Conversely, if det(A) = 0, then the matrix A is singular and does not have an inverse. This is because the determinant being zero implies that the matrix transformation represented by A is not one-to-one, and therefore, the matrix is not invertible.
• Describe how the determinant of a matrix is related to the volume of the parallelotope spanned by its column (or row) vectors.
  • The determinant of a matrix A is related to the volume of the parallelotope spanned by the column (or row) vectors of A in n-dimensional space. Specifically, the absolute value of the determinant, |det(A)|, is equal to the volume of the parallelotope. This means that if the column (or row) vectors of A are linearly independent, then the determinant of A is non-zero, and the volume of the parallelotope is non-zero. Conversely, if the determinant of A is zero, then the column (or row) vectors of A are linearly dependent, and the parallelotope has zero volume.
• Analyze how the determinant of a matrix is affected by elementary row (or column) operations, and explain the significance of this property.
  • The determinant of a matrix A is a multilinear function of the rows (or columns) of the matrix, meaning that it is linear in each row (or column), and the determinant changes linearly when a row (or column) is multiplied by a scalar. This property has significant implications for the behavior of the determinant under elementary row (or column) operations, such as row (or column) swapping, row (or column) scaling, and row (or column) addition. Specifically, swapping two rows (or columns) changes the sign of the determinant, scaling a row (or column) by a scalar multiplies the determinant by that scalar, and adding a multiple of one row (or column) to another does not change the determinant. These properties of the determinant under elementary operations are crucial in various linear algebra techniques, such as finding the inverse of a matrix or solving systems of linear equations using Gaussian elimination.

Related terms

Matrix Inverse: The inverse of a square matrix A, denoted as A^-1, is a matrix that, when multiplied by A, results in the identity matrix. The existence of the matrix inverse is closely related to the determinant of A.

Rank of a Matrix: The rank of a matrix A is the dimension of the column space (or row space) of A, which is equal to the number of linearly independent columns (or rows) of A. The rank of A is closely related to the determinant of A.

Eigenvalues and Eigenvectors: The eigenvalues of a square matrix A are the scalar values $\lambda$ that satisfy the equation Av = $\lambda$v, where v is a non-zero vector. The determinant of A is related to the product of its eigenvalues.