Linear Algebra

Study of vectors, matrices, and linear transformations.

Vectors and Their OperationsMatrices and Matrix ArithmeticLinear Transformations
Determinants and InversesEigenvalues and EigenvectorsVector Spaces and Subspaces
Computer Graphics and AnimationData Science and Machine LearningRobotics and Engineering
Breaking Down Word ProblemsChecking Your Solutions with InversesVisualizing Problems to Find Solutions
Vectors and Their Op...Matrices and Matrix ...Linear Transformatio...Determinants and Inv...Eigenvalues and Eige...Vector Spaces and Su...Computer Graphics an...Data Science and Mac...Robotics and Enginee...Breaking Down Word P...Checking Your Soluti...Visualizing Problems...
Advanced Topics

Determinants and Inverses

Understanding Determinants

The determinant is a special number that tells you important properties about a square matrix, like whether it can be inverted or how it changes area or volume.

  • If the determinant is zero, the matrix squishes space into a lower dimension.
  • If the determinant is not zero, the matrix has an inverse.

Finding the Inverse

The inverse of a matrix is like dividing by a number. If \( A \) is a matrix, its inverse \( A^{-1} \) undoes the transformation of \( A \):

\[ A A^{-1} = I \]

Why Do Determinants Matter?

They help in solving equations and tell us if a system has a unique solution.

Try These

  • Compute the determinant of \[ \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix} \]
  • Find the inverse of a \( 2 \times 2 \) matrix.

Key Formula

\[\det(A) = ad - bc\]

Examples

  • A matrix with determinant zero cannot be inverted.

  • The inverse of a \( 2 \times 2 \) matrix can be found using the determinant.

In a Nutshell

Determinants tell us important things about matrices, and inverses help us undo matrix transformations.

Key Terms

Determinant
A number that reveals properties like invertibility of a matrix.
Inverse Matrix
A matrix that reverses the effect of the original matrix.
Next
Eigenvalues and Eigenvectors