Determinants

Definition

Determinants are a numerical value associated with a square matrix that provides important information about the matrix, such as whether it is invertible or has a unique solution. Determinants play a crucial role in solving systems of linear equations, particularly those involving three variables.

5 Must Know Facts For Your Next Test

1. The determinant of a 2x2 matrix is calculated as ad - bc, where a, b, c, and d are the elements of the matrix.
2. For a 3x3 matrix, the determinant 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 elements of the matrix.
3. If the determinant of a matrix is zero, the matrix is not invertible, and the system of equations represented by the matrix may have no unique solution or may have an infinite number of solutions.
4. The determinant of a matrix can be used to calculate the area of the parallelogram formed by the column (or row) vectors of the matrix.
5. Cofactors are used in the calculation of determinants and can also be used to find the inverse of a matrix.

Review Questions

• Explain how the determinant of a 3x3 matrix is calculated and what this value represents.
  • The determinant of a 3x3 matrix is 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 elements of the matrix. This value represents important information about the matrix, such as whether it is invertible or has a unique solution. If the determinant is zero, the matrix is not invertible, and the system of equations represented by the matrix may have no unique solution or may have an infinite number of solutions.
• Describe the relationship between the determinant of a matrix and the invertibility of the matrix.
  • The determinant of a matrix is directly related to the invertibility of the matrix. If the determinant of a matrix is non-zero, the matrix is invertible, meaning that 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, the matrix is not invertible, and the system of equations represented by the matrix may have no unique solution or may have an infinite number of solutions.
• Explain how the determinant of a matrix can be used to calculate the area of the parallelogram formed by the column (or row) vectors of the matrix.
  • The determinant of a matrix can be used to calculate the area of the parallelogram formed by the column (or row) vectors of the matrix. Specifically, the absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column (or row) vectors of the matrix. For a 3x3 matrix, the absolute value of the determinant represents the volume of the parallelepiped formed by the column (or row) vectors of the matrix. This property of determinants is useful in various applications, such as in linear algebra, geometry, and physics.

Related terms

Matrix: A rectangular array of numbers, symbols, or expressions, arranged in rows and columns, that can be used to represent and manipulate mathematical relationships.

Inverse Matrix: A matrix that, when multiplied by a given matrix, results in the identity matrix, indicating that the original matrix is invertible.

Cofactor: A scalar value obtained by removing a row and column from a matrix and calculating the determinant of the resulting smaller matrix.