Singular Matrix and Its Properties
A singular matrix is a square matrix that does not have an inverse. Mathematically, a matrix \( A \) is said to be singular if its determinant is zero:
\[ \det(A) = 0 \]
Since the inverse of a matrix \( A \) is given by:
\[ A^{-1} = \frac{1}{\det(A)} \text{ adj}(A) \]
if \( \det(A) = 0 \), the denominator is zero, making \( A^{-1} \) undefined. This means a singular matrix is not invertible.
Properties of Singular Matrices
1. Determinant is Zero
The most fundamental property of a singular matrix is that its determinant is zero:
\[ \det(A) = 0 \]
Example:
Consider the following \( 2 \times 2 \) matrix:
\[ A = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \]
Its determinant is:
\[ \det(A) = (2 \times 2) – (4 \times 1) = 4 – 4 = 0 \]
Since the determinant is zero, \( A \) is singular.
2. No Inverse Exists
Since a singular matrix has a determinant of zero, it is not invertible.
Example:
For the matrix:
\[ B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \]
The determinant is:
\[ \det(B) = (1 \times 4) – (2 \times 2) = 4 – 4 = 0 \]
Since \( B \) is singular, it does not have an inverse.
3. Linearly Dependent Rows or Columns
A matrix is singular if one row (or column) is a scalar multiple of another, making the rows or columns linearly dependent.
Example:
In the matrix:
\[ C = \begin{bmatrix} 3 & 6 \\ 1 & 2 \end{bmatrix} \]
The second row is a multiple of the first row:
\[ \text{Row 2} = \frac{1}{3} \times \text{Row 1} \]
Since the rows are linearly dependent, \( C \) is singular.
4. Zero Eigenvalue
A singular matrix always has at least one eigenvalue equal to zero.
Example:
For the matrix:
\[ D = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \]
The characteristic equation is found by solving:
\[ \det(D – \lambda I) = 0 \]
\[ \begin{vmatrix} 2-\lambda & 4 \\ 1 & 2-\lambda \end{vmatrix} = 0 \]
Expanding:
\[ (2-\lambda)(2-\lambda) – (4 \times 1) = \lambda^2 – 4\lambda = 0 \]
Solving for \( \lambda \):
\[ \lambda(\lambda – 4) = 0 \]
\[ \lambda = 0, 4 \]
Since one eigenvalue is zero, \( D \) is singular.
5. Not Full Rank
A singular matrix has a rank lower than its order. That is, an \( n \times n \) singular matrix has rank \( r < n \).
Example:
For the matrix:
\[ E = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix} \]
Each row is a multiple of the first row, so there is only one independent row. The rank is 1, which is less than 3, making \( E \) singular.