Properties of determinants

Definition

Determinants are scalar values that can be computed from the elements of a square matrix. They play a crucial role in solving systems of linear equations, particularly when using techniques like 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 given by $ad - bc$.
2. If the determinant of a matrix is zero, the system of equations has no unique solution or is dependent.
3. For an $n \times n$ matrix, swapping two rows (or columns) changes the sign of the determinant.
4. Multiplying a row (or column) by a scalar multiplies the determinant by that scalar.
5. The determinant of an identity matrix is always one.

Review Questions

• How do you compute the determinant of a $3 \times 3$ matrix?
• What does it mean if the determinant of a matrix is zero?
• Describe how row operations affect the value of a determinant.

Related terms

Matrix: A rectangular array of numbers arranged in rows and columns used to represent linear transformations and solve systems of linear equations.

Cramer's Rule: A theorem used to solve systems of linear equations with as many equations as unknowns using determinants.

Inverse Matrix: A matrix which, when multiplied by its original matrix, results in an identity matrix. It exists only if the determinant is non-zero.