Determinants Reminder

What is the determinant of a square matrix?

The determinant is a scalar value computed from a square matrix A\in\mathbb{F}^{n\times n} that encodes important properties such as invertibility, volume scaling, and orientation. It is denoted det(A) or |A|.

When is a matrix invertible in terms of its determinant?

A matrix A is invertible iff det(A)\neq0. If det(A)=0, the matrix is singular and has no inverse.

What is the determinant of a triangular matrix?

For any upper or lower triangular matrix, the determinant is the product of the diagonal entries.

What effect does a row operation have on the determinant?

• Swapping rows \rightarrow changes sign of the determinant

• Multiplying a row by c\rightarrow multiplies determinant by c

• Adding a multiple of one row to another \rightarrow no change in determinant

How is the determinant used in geometry?

The determinant of a matrix columns are vectors gives the oriented volume of the parallelepiped formed by the vectors. In \mathbb{R}², it gives signed area; in \mathbb{R}³, signed volume.

What is the determinant of a 2\times2 matrix?

det\left[\begin{array}{cc} a&b\\c&d\end{array}\right]=ad-bc

What is the determinant of a 3\times 3 matric?

Use cofactor expansion det(A)=a(ei-fh)-b(di-fg)+c(dh-eg) for A=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right]

How do you compute the determinant of an n\times n matrix?

1. Use cofactor expansion along a row or column

2. For larger matrices, reduce to triangular form and multiply diagonal.

3. Be cautious of row operation effects on determinant

What’s the relationship between determinant and eigenvalues?

det(A)=\prod^n_{i=0}\lambda_i where \lambda_i are the eigenvalues (with multiplicity) of A.