Determinant

Definition

The determinant of a square matrix is a scalar value that is a function of the entries of the matrix. It carries important information about the matrix, such as whether the matrix is invertible and the volume of the parallelepiped spanned by the column vectors of the matrix.

5 Must Know Facts For Your Next Test

1. The determinant of a $2\times 2$ matrix is calculated as $ad - bc$, where $a$, $b$, $c$, and $d$ are the entries of the matrix.
2. The determinant of a $3\times 3$ matrix can be calculated using the formula $a(ei - fh) - b(di - fg) + c(dh - eg)$, where $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$, and $i$ are the entries of the matrix.
3. The determinant of a matrix is a scalar value that can be positive, negative, or zero.
4. If the determinant of a square matrix is non-zero, then the matrix is invertible. If the determinant is zero, then the matrix is not invertible.
5. Cramer's Rule uses the determinant of the coefficient matrix and the matrices formed by replacing the columns of the coefficient matrix with the constant terms to solve systems of linear equations.

Review Questions

• Explain how the determinant of a matrix is related to the invertibility of the matrix.
  • The determinant of a square matrix is a scalar value that provides important information about the matrix. If the determinant of a matrix is non-zero, then the matrix is invertible, meaning there exists a unique inverse matrix that, when multiplied by the original matrix, results in the identity matrix. Conversely, if the determinant of a matrix is zero, then the matrix is not invertible, and there is no unique inverse matrix that can be found.
• Describe how the determinant of a matrix is used in Cramer's Rule to solve systems of linear equations.
  • Cramer's Rule is a method for solving systems of linear equations that utilizes the determinant of the coefficient matrix and the determinants of the matrices formed by replacing the columns of the coefficient matrix with the constant terms. Specifically, the solution for each variable in the system is given by the ratio of the determinant of the matrix formed by replacing the corresponding column of the coefficient matrix with the constant terms, divided by the determinant of the coefficient matrix. This approach allows for the efficient computation of the solution to the system of linear equations using determinants.
• Analyze the relationship between the volume of the parallelepiped spanned by the column vectors of a matrix and the determinant of that matrix.
  • The determinant of a square matrix is closely related to the volume of the parallelepiped spanned by the column vectors of that matrix. Specifically, the absolute value of the determinant is equal to the volume of the parallelepiped. This means that if the determinant of a matrix is zero, the column vectors of the matrix are linearly dependent, and the parallelepiped has zero volume. Conversely, if the determinant is non-zero, the column vectors are linearly independent, and the parallelepiped has a non-zero volume. This connection between the determinant and the geometry of the matrix provides valuable insights into the properties and applications of determinants.

Related terms

Matrix: A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Inverse Matrix: A matrix that, when multiplied by another matrix, results in the identity matrix.

Cramer's Rule: A method for solving systems of linear equations by using the determinants of the coefficient matrix and the matrices formed by replacing the columns of the coefficient matrix with the constant terms.