Det(A)

Definition

The determinant of a square matrix A, denoted as det(A), is a scalar value that provides important information about the matrix, such as whether it is invertible and the volume of the parallelotope formed by its column (or row) vectors. The determinant is a fundamental concept in linear algebra with applications in various fields, including physics, economics, and computer science.

5 Must Know Facts For Your Next Test

1. The determinant of a $2\times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is given by the formula $ad - bc$.
2. If the determinant of a square matrix A is zero, then A is a singular matrix and does not have a matrix inverse.
3. The determinant of a matrix is invariant under row or column operations, meaning that performing elementary row or column operations on a matrix does not change its determinant.
4. The determinant of a diagonal matrix is the product of its diagonal elements.
5. The determinant of a triangular matrix is the product of its diagonal elements.

Review Questions

• Explain the relationship between the determinant of a matrix and the invertibility of the matrix.
  • The determinant of a square matrix A, denoted as det(A), is closely related to the invertibility of the matrix. If det(A) is non-zero, then the matrix A is invertible, meaning it has a matrix inverse, A^(-1), that satisfies the equation A * A^(-1) = A^(-1) * A = I, where I is the identity matrix. Conversely, if det(A) is zero, then the matrix A is singular and does not have a matrix inverse. This is because the determinant being zero implies that the matrix is not full rank, and therefore, it cannot be invertible.
• Describe how the determinant of a matrix can be used to compute the volume of the parallelotope formed by the column (or row) vectors of the matrix.
  • The determinant of a square matrix A can be interpreted as the signed volume of the parallelotope formed by the column (or row) vectors of the matrix. Specifically, if A is an $n\times n$ matrix, then the absolute value of det(A) is equal to the volume of the $n$-dimensional parallelotope spanned by the column (or row) vectors of A. The sign of the determinant indicates the orientation of the parallelotope, with a positive sign indicating a right-handed orientation and a negative sign indicating a left-handed orientation.
• Explain how the determinant of a matrix can be used to solve systems of linear equations using the inverse matrix method.
  • The determinant of a matrix A plays a crucial role in the inverse matrix method for solving systems of linear equations of the form Ax = b, where A is a square matrix and b is a vector of constants. If the determinant of A is non-zero, then A is invertible, and the unique solution to the system is given by x = A^(-1)b. The existence of the matrix inverse A^(-1) is guaranteed if and only if the determinant of A is non-zero. Therefore, the determinant of the coefficient matrix A is an important factor in determining the solvability and uniqueness of the solution to the system of linear equations.

Related terms

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

Cofactor Expansion: A method for computing the determinant of a square matrix by expanding along a row or column, using the cofactors of the elements in that row or column.

Singular Matrix: A square matrix A is said to be singular if its determinant is zero, meaning it does not have a matrix inverse. A non-singular matrix has a non-zero determinant and is invertible.