A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties about the matrix such as whether it is invertible.
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The determinant of a 2x2 matrix $\begin{pmatrix}a & b\\c & d\end{pmatrix}$ is calculated as $ad - bc$.
The determinant helps determine if a system of linear equations has a unique solution; if the determinant is zero, the system does not have a unique solution.
In Cramer's Rule, determinants are used to solve linear systems by expressing solutions in terms of ratios of determinants.
The determinant of an upper or lower triangular matrix is the product of its diagonal elements.
Swapping two rows (or columns) of a matrix multiplies its determinant by -1.
Review Questions
What is the determinant of the 2x2 matrix $\begin{pmatrix}3 & 8\\4 & 6\end{pmatrix}$?
How can you use determinants to determine if a system of equations has no unique solution?
If you swap two rows in a matrix, how does that affect the determinant?