Inverse matrix

Definition

An inverse matrix is a matrix that, when multiplied by its original matrix, yields the identity matrix. It is denoted as $A^{-1}$ for a given matrix $A$.

5 Must Know Facts For Your Next Test

1. The inverse of a matrix exists only if the determinant of the matrix is non-zero.
2. For a 2x2 matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the inverse is calculated as $\frac{1}{ad-bc}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$.
3. If $A$ and $B$ are invertible matrices, then $(AB)^{-1} = B^{-1}A^{-1}$.
4. The product of a matrix and its inverse results in the identity matrix, i.e., $AA^{-1} = I$.
5. Inverting larger matrices often involves row reduction or using software tools like MATLAB or Python.

Review Questions

• What condition must be met for a square matrix to have an inverse?
• How do you find the inverse of a 2x2 matrix?
• What is the result of multiplying a matrix by its inverse?

Related terms

Determinant:

A scalar value that is computed from the elements of a square matrix and determines whether the matrix has an inverse.

Identity Matrix:

A square matrix with ones on the diagonal and zeros elsewhere; it acts as the multiplicative identity in matrix multiplication.

Row Reduction:

A method used to simplify matrices to their row echelon form to solve systems of linear equations or find inverses.