Singular Matrix and Its Properties

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:

$$A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$$

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:

$$B=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$$

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:

$$C=\left[\begin{array}{cc}3& 6\\ 1& 2\end{array}\right]$$

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:

$$D=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$$

The characteristic equation is found by solving:

$$det(D–\lambda I)=0$$

$$\left|\begin{array}{cc}2-\lambda & 4\\ 1& 2-\lambda \end{array}\right|=0$$

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:

$$E=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$$

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.