Non-Singular Matrix

In linear algebra, matrices are the most essential part of higher math, especially when getting ready for the JEE. Non-singular matrices are necessary for solving equations, figuring out if something can be inverted, and looking at systems of linear equations.

A non-singular matrix is one whose determinant is not zero. This property sets it apart from a singular matrix, which has a determinant that equals zero. To do well on competitive tests like the JEE, you need to know the non-singular matrix formula, its properties, and some examples.

1.0What Is a Non-Singular Matrix?

A non-singular matrix is a square matrix (the number of rows equals the number of columns) that has a non-zero determinant. In mathematical terms, if ( A ) is a square matrix and ( $\text{det}(A)\ne 0$ ), then ( A ) is non-singular.

Condition: Only square matrices (order n×n) can be non-singular.

Key Insight: A non-singular matrix always has an inverse, which is why it is sometimes called an invertible matrix.

2.0Non-Singular Matrix Formula

The most crucial formula related to non-singular matrices is the inverse formula.

If A is a non-singular matrix of order n, its inverse is given by:

$A^{-1}=\frac{Adj(A)}{det(A)}$

Where:

• Adj(A) = Adjoint of the matrix A
• det(A) ≠ 0

This formula highlights the fact that an inverse exists only when the matrix is non-singular.

3.0How To Find A Non-Singular Matrix?

The determinant test is the main way we find out if a matrix is non-singular. This process is simple and follows a set pattern because the determinant is what matters. Let's take it apart:

Step 1: Check if the Matrix is Square

• Only square matrices (matrices with an equal number of rows and columns) can be classified as singular or non-singular.
• If the matrix is not square, the concept of non-singular does not apply.

Step 2: Calculate the Determinant

• For a 2 × 2 matrix: $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$\det (A)=ad-bc$
• For a 3 × 3 matrix: $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$= $\det (A)=a(ei-fh)-b(di-fg)+c(dh-eg)$
• For higher-order matrices, determinants are calculated using expansion by minors or row-reduction methods.

Step 3: Apply the Determinant Condition

• If det⁡(A)=0, then the matrix is singular.
• If det⁡(A)≠0, then the matrix is non-singular.

Step 4: Verify with Rank (Optional for JEE-level cross-check)

• A square matrix of order n is non-singular if its rank = n.
• This means all rows (or columns) are linearly independent.

Step 5: Check for Inverse (Alternative Method)

• If you can compute an inverse for the matrix using: $A^{-1}=\frac{Adj(A)}{det(A)}$, then A must be non-singular, since this formula is valid only when det⁡(A)≠0.

4.0Rules For Row and Column Operations of a Determinant

While solving determinants, especially for larger matrices, row and column operations simplify the process. But certain rules must be followed carefully:

1. Swapping Rows or Columns: If any two rows (or columns) are interchanged, the sign of the determinant changes.

• $det(A)\to -det(A)$

2. Identical Rows or Columns: The determinant is zero when two rows (or columns) are either the same, or proportional.
3. Row/Column Multiplication: When you multiply a row or column (of any matrix) by a scalar k, the determinant is multiplied by k (the same scalar).
4. Row/Column Addition Rule: Adding a multiple of one row to another (or a column to another) does not affect the value of the determinant. This property will be used to simplify matrices before finding a determinant.
5. Zero Row or Column: If any row (or column) is completely zero, the determinant is zero and the matrix is singular.

5.0Properties of Non-Singular Matrix

Understanding the non-singular matrix properties is vital for JEE and advanced mathematics:

1. Non-Zero Determinant: The determinant of a matrix is never zero ($(\text{det}(A)\ne 0)$).
2. Existence of Inverse: Only non-singular matrices have an inverse $(A^{-1}))$, such that $(A{A}^{-1}=A^{-1}A=I)$ where (I) is the identity matrix.
3. Unique Solutions: In the equation $(A\vec{x}=\vec{b})$, if (A) is non-singular, a unique solution $(\vec{x})$ exists.
4. Product of Non-Singular Matrices: The product of two non-singular matrices is also non-singular.$(\text{det}(AB)=\text{det}(A)\cdot \text{det}(B)\ne 0)$
5. Row Operations: Non-singularity is preserved under elementary row operations (except those that make the determinant zero).
6. Transpose is Non-Singular: The transpose of a non-singular matrix is also non-singular.
7. All Eigenvalues Are Non-Zero: A non-singular matrix has no zero eigenvalues.

6.0Examples on Non-Singular Matrix

Example 1: 2x2 Matrix

$A=\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$

$\text{det}(A)=(4\times 6)-(7\times 2)=24-14=10\ne 0$

Conclusion: (A) is a non-singular matrix.

Example 2: 3x3 Matrix

$B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 5& 6& 0\end{array}\right]$

$$\begin{gathered} \text{det}(B)=1(1\times 0-4\times 6)-2(0\times 0-4\times 5)+3(0\times 6-1\times 5) \\ =1(0-24)-2(0-20)+3(0-5) \\ =-24+40-15=1 \end{gathered}$$

Conclusion: (B) is a non-singular matrix.

Example 3: Singular Matrix (for Comparison)

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

$\text{det}(C)=(2\times 2)-(4\times 1)=4-4=0$

Conclusion: (C) is a singular matrix (not non-singular).

7.0Practice Questions on Non-Singular Matrix

1. Determine if the following matrix is non-singular: $\left[\begin{array}{cc}5& 8\\ 2& 3\end{array}\right]$
2. Find the inverse of the following non-singular matrix:$\left[\begin{array}{cc}2& 1\\ 7& 4\end{array}\right]$
3. Is the matrix below non-singular or singular? Justify your answer with a calculation: $\left[\begin{array}{ccc}0& 1& 2\\ 1& 0& 3\\ 4& 5& 6\end{array}\right]$
4. If (A) and (B) are both non-singular matrices, prove that (AB) is also non-singular.
5. If the determinant of a matrix (A) is 0, what can you say about the singularity of (A)?