๐ What is an Invertible Matrix?
An invertible matrix, also known as a non-singular matrix, is a square matrix for which there exists another matrix (its inverse) that, when multiplied with it, results in the identity matrix. In simpler terms, it's a matrix that can be 'undone'.
๐ค What is a Singular Matrix?
A singular matrix, on the other hand, is a square matrix that does not have an inverse. This typically occurs when the determinant of the matrix is zero. It cannot be 'undone'.
๐ Invertible vs. Singular Matrices: A Detailed Comparison
| Feature |
Invertible Matrix |
Singular Matrix |
| Definition |
A square matrix that has an inverse. |
A square matrix that does not have an inverse. |
| Determinant |
The determinant is non-zero ($det(A) \neq 0$). |
The determinant is zero ($det(A) = 0$). |
| Inverse |
Has an inverse matrix, denoted as $A^{-1}$. |
Does not have an inverse. |
| Linear Independence |
Columns (and rows) are linearly independent. |
Columns (and rows) are linearly dependent. |
| Solutions to $Ax = b$ |
Has a unique solution for every $b$. |
May have no solution or infinitely many solutions for a given $b$. |
| Rank |
Full rank (equal to the number of rows/columns). |
Less than full rank. |
| Eigenvalues |
All eigenvalues are non-zero. |
At least one eigenvalue is zero. |
๐ก Key Takeaways
- ๐ข Determinant: An invertible matrix has a non-zero determinant, while a singular matrix has a zero determinant.
- ๐๏ธ Inverse Existence: Only invertible matrices possess an inverse, allowing for the 'undoing' of transformations.
- ๐ Linear Independence: Invertible matrices have linearly independent rows and columns, ensuring a unique solution to linear equations.
- ๐ซ Singular Matrix Solutions: Singular matrices lead to either no solution or infinite solutions for linear equations.
- ๐ Rank Implications: The rank of an invertible matrix is full, indicating its non-degeneracy, whereas a singular matrix has a reduced rank.
- ๐งฎ Eigenvalue Significance: All eigenvalues of an invertible matrix are non-zero, contrasting with singular matrices which have at least one zero eigenvalue.
