4 Determinants Free Online FlashCards

What kind of quantity is a determinant?

A determinant is a scalar associated with a square matrix; it is defined only for square matrices.

What is the determinant formula for a $2\times2$ matrix?

For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, $\det(A)=ad-bc$.

What does a row swap do to a determinant?

Interchanging two rows changes the sign of the determinant.

What does row replacement do to a determinant?

Adding a multiple of one row to another row leaves the determinant unchanged.

How is the determinant of a triangular matrix computed?

The determinant is the product of the diagonal entries: $\det(A)=a_{11}a_{22}\cdots a_{nn}$.

What is the determinant product rule?

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

How does multiplying an entire matrix by $k$ affect its determinant?

For an $n\times n$ matrix, $\det(kA)=k^n\det(A)$.

What does the absolute determinant measure geometrically?

The absolute value $|\det(A)|$ is the factor by which the transformation scales $n$-dimensional volume.

What does a negative determinant indicate?

A negative determinant means the transformation reverses orientation while scaling volume by $|\det(A)|$.

What does $\det(A)=0$ mean geometrically?

A zero determinant means the transformation collapses space into a lower-dimensional set, so the matrix is not invertible.

How are a minor and its cofactor related?

The minor $M_{ij}$ is the determinant after deleting row $i$ and column $j$; the cofactor is $C_{ij}=(-1)^{i+j}M_{ij}$.

What is cofactor expansion along row $i$?

Expanding along row $i$ gives $\det(A)=\sum_{j=1}^{n}a_{ij}C_{ij}$; the analogous column formula also works.