Properties of determinants

Definition

The determinant is a scalar value that is computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. It is crucial for solving systems of linear equations using Cramer's Rule.

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 computed as $ad - bc$.
2. A matrix with a zero determinant is singular, meaning it does not have an inverse.
3. Swapping two rows or columns of a matrix multiplies its determinant by $-1$.
4. Multiplying a row or column of a matrix by a scalar multiplies the determinant by that scalar.
5. The determinant of the product of two matrices equals the product of their determinants, i.e., $\det(AB) = \det(A) \cdot \det(B)$.

Review Questions

• How do you compute the determinant of a $2 \times 2$ matrix?
• What happens to the determinant if you swap two rows in a matrix?
• Why is it important to know whether a matrix has a zero determinant when solving systems of equations?

Related terms

Cramer's Rule: A method used to solve systems of linear equations with as many equations as unknowns, utilizing determinants.

Singular Matrix: A square matrix that does not have an inverse, indicated by its zero determinant.

Matrix Inverse: \(A^{-1}\), which exists only for non-singular matrices, satisfying \(AA^{-1} = I\).