Finding Inverses of Square Matrices
A square matrix has a multiplicative inverse if and only if it can be reduced to the identity matrix using elementary row operations.
The Inverse Matrix Concept
For a square matrix A, its multiplicative inverse is a matrix denoted A inverse such that their product in either order yields the identity matrix. If such a matrix exists, A is called invertible or non-singular. If no inverse exists, A is called singular.
A A^{-1} = A^{-1} A = I_nGauss-Jordan Elimination Algorithm
To compute the inverse of an n-by-n matrix A, construct an augmented matrix with A on the left and the identity matrix on the right. Applying row operations to reach reduced row echelon form transforms the identity block into A inverse. If the left side cannot become the identity matrix, then A is singular.
[A \mid I_n] \longrightarrow [I_n \mid A^{-1}]Explicit Formula for 2x2 Matrices
For a 2-by-2 matrix, the inverse can be computed directly using its determinant. Swap the main diagonal entries, negate the off-diagonal entries, and divide the matrix by ad minus bc. The matrix is invertible if and only if this determinant value is non-zero.
\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}Flashcard
The unique matrix satisfying matrix multiplication to the identity.
Check your understanding
Which condition guarantees that a square matrix A is invertible?