Singular Matrix

In linear algebra, matrices play a crucial role in solving equations, performing transformations, and modelling real-life structures. Singularity of matrices is a major classification.

Singular matrices commonly arise in determinant problems dealing with inverse matrices. JEE aspirants will have to understand the definition, formula, properties, and worked examples of singular matrices to be able to tackle higher-level questions.

Singular matrices are ideas that flow from determinants and serve as the basis for prospective authors wishing to solve problems involving matrices in exams.

1.0What is a Singular Matrix?

A square matrix is called singular if its determinant is 0. This is one of the least difficult, yet most important, kinds of matrices found in mathematics. The word "singular" means simply that the matrix cannot be inverted.

2.0Singular Matrix Formula

If A is an $n\times n$ matrix, then:

$\text{If}|A|=0,\text{then}A\text{ is a singular matrix.}$

Here, |A| denotes the determinant of matrix A.

It's important to know when and why a matrix is singular in order to answer questions about determinants, systems of linear equations, and matrix inversion.

Singular Matrix Example:

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

$det(A)=(2)(2)-(4)(1)=4-4=0$

Thus, A is a singular matrix.

3.0Identifying a Singular Matrix

Steps to Identify a Singular Matrix

1. Check if the Matrix is Square: Only square matrices (same number of rows and columns) can be singular.
2. Calculate the Determinant:

• If |A| = 0, then A is a singular matrix.
• If $|A|\ne 0$, then A is a non-singular matrix.

3. Observe Linear Dependence: If any row or column is a linear combination of others, the determinant will be zero, making it singular.

Singular Matrix Determinant

• For $2\times 2$ matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$: |A| = ad - bc
• For $3\times 3$ matrix $A=\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$: |A| = a(ei - fh) - b(di - fg) + c(dh - eg)

4.0Properties of Singular Matrix

To understand how singular matrices work and what they mean in algebra and geometry, you need to know their properties.

Key Singular Matrix Properties

• Zero Determinant: The most important property to know is that |A| = 0.
• There is no inverse: Singular matrices don't have an inverse. Only matrices with |A| not equal to 0 can be inverted.
• Linear Dependence: One row or column is a linear combination of the others.
• Rank is Less Than Order: The rank of a single matrix is always lower than the number of its rows or columns.
• Linear Systems with Non-Unique Solutions: If the coefficient matrix in a system of equations is singular, the system has either no solution or an infinite number of solutions, but never just one solution.
• Determinant Formula for a Fast Check: For quick identification, use the singular matrix formula |A|=0.

Learn More: Adjoint of a Matrix

5.0Difference Between Singular and Non-Singular Matrix

Understanding the difference is crucial for JEE-level problem solving.

6.0Theorem to Generate Singular Matrices

Linear Dependence Theorem

Theorem: If any row (or column) of a square matrix is a linear combination of other rows (or columns), then the matrix is singular.

Proof Outline: If, say, R2 = kR1, then after performing row operations, rows become linearly dependent, making the determinant zero.

Direct Construction:

• Make two rows (or columns) proportional.
• Make one row/column the sum or difference of other rows/columns.

Example using Theorem:

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

Row 2 = 2 × Row 1, so |A| = 0 and A is singular.

7.0Solved Questions on Singular Matrix

Solved Example 1: Finding Value for Singularity

Question: For what value of x will $A=\left[\begin{array}{ccc}x& 23& 6\end{array}\right]$ be singular?

Solution:

• Determinant: $x\times 6-2\times 3=6x-6$
• Set to zero: $6x-6=0⟹x=1$

Solved Example 2: Checking for Singularity

Question: Is $B=\left[\begin{array}{cc}2& 5\\ 4& 10\end{array}\right]$ a singular matrix?

Solution:

• Determinant: $2\times 10-5\times 4=20-20=0$
• Yes, B is singular.

Solved Example 3: 3x3 Matrix

Question: Show that $C=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ is singular.

Solution:

Determinant:

$=1(5\times 9-6\times 8)-2(4\times 9-6\times 7)+3(4\times 8-5\times 7)$

$=1(45-48)-2(36-42)+3(32-35)$

$=-3+12-9=0$

So, C is singular.

Solved Example 4: Finding Parameter Value

Question: Find k if $D=\left(\begin{array}{cc}k& 1\\ 2& 2\end{array}\right)$ is singular.

Solution:

• Determinant: $k\times 2-1\times 2=2k-2$
• Set to zero: $2k-2=0⟹k=1$

8.0Practice Questions on Singular Matrix

1. For which value of a is $A=\left(\begin{array}{ccc}a& 32& 6\end{array}\right)$ singular?
2. Is $B=\left(\begin{array}{cc}0& 4\\ 0& 8\end{array}\right)$ a singular matrix?
3. If $C=\left(\begin{array}{cc}5& 7\\ 15& k\end{array}\right)$ is singular, find k.
4. Construct a $3\times 3$ singular matrix other than those given in examples above.
5. If $D=\left(\begin{array}{cc}2& x\\ 8& 4\end{array}\right)$ is singular, find x.
6. Is the matrix $E=\left(\begin{array}{ccc}2& 4& 6\\ 1& 2& 3\\ 3& 6& 9\end{array}\right)$ singular?