Determinants: Properties, Cofactors, Computation, and Applications

All flashcards

What is a determinant?

A scalar associated with a square matrix that encodes properties such as invertibility and volume scaling.

What is the determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

$ad-bc$.

What does the absolute determinant of a real square matrix measure?

The factor by which its linear transformation scales full-dimensional volume.

What does the sign of a nonzero determinant indicate?

Whether the associated linear transformation preserves or reverses orientation.

What is the determinant product rule?

For square matrices of the same size, $\det(AB)=\det(A)\det(B)$.

Quiz questions

1. Which matrices have determinants?Answer: Square matrices.
2. What happens to a determinant when a multiple of one row is added to another row?Answer: The determinant remains unchanged.
3. Over a field, when is a square matrix invertible?Answer: Exactly when its determinant is nonzero.
4. For square matrices $A$ and $B$ of the same size, which identity holds?Answer: $\det(AB)=\det(A)\det(B)$
5. How are the eigenvalues of a square matrix $A$ related to its characteristic polynomial $\det(XI-A)$?Answer: They are the roots of that polynomial.