# Non-Singular Matrix — Definition, Properties, and Examples

## What Is a Non-Singular Matrix?

A **non-singular matrix** is a square matrix AAA whose determinant is not equal to zero, that is, \(\det A \neq 0\). Because its determinant is non-zero, a non-singular matrix always has an **inverse**, which is why it is also called an **invertible matrix**. Only square matrices (order \(n \times n\)) can be singular or non-singular, since only square matrices have a determinant.

> **Quick Reference:**  
> **Definition:** A square matrix AAA with \(\det A \neq 0\).  
> **Also called:** invertible matrix  
> **Condition:** \(\det A \neq 0\) (equivalently, rank = n; rows/columns linearly independent)  
> **Type:** property of a square matrix  
> **Used in:** solving linear systems, matrix inverses, computer graphics, machine learning

The two tests below always agree, so you can use whichever is easier for the matrix in front of you:

- **Determinant test:** \(\det A \neq 0 \Rightarrow \) non-singular.
- **Rank test:** a matrix of order nnn is non-singular if and only if its rank equals nnn, so every row (and column) is linearly independent.

## Non-Singular vs. Singular Matrix

The cleanest way to hold the definition is by its opposite. A **singular matrix** has \(\det A = 0\); it is _not_ invertible, and its rows or columns are linearly dependent.

| Feature | Non-singular matrix | Singular matrix |
| --- | --- | --- |
| Determinant | \(\det A \neq 0\) | \(\det A = 0\) |
| Inverse | exists (invertible) | does not exist |
| Rank (order n) | rank = n | rank < n |
| Rows / columns | linearly independent | linearly dependent |
| System AX=B | unique solution | no unique solution |

## How to Check If a Matrix Is Non-Singular

The procedure is short:

1. Confirm the matrix is square (equal number of rows and columns). A non-square matrix is neither singular nor non-singular.
2. Compute the determinant.
3. If the determinant is non-zero, the matrix is non-singular; if it is zero, the matrix is singular.

## Properties of Non-Singular Matrices

- The **product** of two non-singular matrices of the same order is non-singular: \(\det(AB) = \det(A)\det(B)\), and neither factor is zero.
- If AAA is non-singular, so is any non-zero scalar multiple \(kA\) (for \(k \neq 0\)).
- The **inverse** \(A^{-1}\) of a non-singular matrix is itself non-singular.
- The **transpose** \(A^{T}\) of a non-singular matrix is non-singular, since \(\det A^{T} = \det A\).
- The **identity matrix** is non-singular, with \(\det I = 1\).

## Examples of Non-Singular Matrix

### Example 1
**Is A=\[1 -4 \ 3 5\] non-singular?**

Apply \(\det A = ad - bc\):
\[
det A = (1)(5) - (-4)(3) = 5 + 12 = 17
\]
Since \(17 \neq 0\), A is non-singular (invertible).
**Final answer:** Non-singular, \(\det A = 17\).

### Example 2
**Is B=\[3 6 \ 2 4\] non-singular?**  
Compute the determinant:
\[
det B = (3)(4) - (6)(2) = 12 - 12 = 0
\]
The determinant is zero, so B is **singular**.  
**Final answer:** Singular, \(\det B = 0\).

### Example 3  
**Is C=\[2 0 \ 0 7\] non-singular?**  
\[
det C = (2)(7) = 14
\]
Since \(14 \neq 0\), C is non-singular.
**Final answer:** Non-singular, \(\det C = 14\).

### Example 4  
**Is D=\[4 -1 0 \ 2 3 5 \ -1 7 2\] non-singular?**  
Expand along the first row:
\[
det D = 4\begin{vmatrix} 3 & 5 \ 7 & 2 \end{vmatrix} + 1\begin{vmatrix} 2 & 5 \ -1 & 2 \end{vmatrix}
\]
Compute each minor:
\[
\begin{vmatrix} 3 & 5 \ 7 & 2 \end{vmatrix} = (3)(2) - (5)(7) = 6 - 35 = -29
\]
\[
\begin{vmatrix} 2 & 5 \ -1 & 2 \end{vmatrix} = (2)(2) - (5)(-1) = 4 + 5 = 9
\]
Substitute:
\[
det D = 4(-29) + 1(9) = -116 + 9 = -107
\]
Since \(-107 \neq 0\), D is non-singular.
**Final answer:** Non-singular, \(\det D = -107\).

### Example 5  
**Find k so that E=\[k 2 \ 3 6\] is singular.**  
Set the determinant to zero:
\[
det E = 6k - 6 = 0 \Rightarrow k = 1
\]
**Final answer:** Singular at \(k = 1\); non-singular for all \(k \neq 1\).

### Example 6
**If A and B are non-singular with \(\det A = 5\) and \(\det B = -2\), is AB non-singular?**
\[
det(AB) = \det(A)\det(B) = (5)(-2) = -10
\]
Since \(-10 \neq 0\), AB is non-singular.
**Final answer:** Non-singular, \(\det(AB) = -10\).

## Why the Non-Singular Property Matters: "Can this be undone?"

- **Solving equations:** A linear system \(AX=B\) has a unique solution \(X=A^{-1}B\) exactly when A is non-singular. 
- **Computer graphics:** rotation and scaling matrices must be non-singular.
- **Data and machine learning:** many algorithms invert a matrix; a singular (or nearly singular) matrix breaks the computation.

## What Are the Most Common Mistakes With Non-Singular Matrices?

### Mistake 1: Judging singularity without computing the determinant
**The correct way:** Always compute \(\det A = ad - bc\).

### Mistake 2: Testing a non-square matrix
**The correct way:** Only square matrices can be singular or non-singular.

### Mistake 3: Confusing zero entries with a zero determinant
**The correct way:** A matrix full of zeros can still have a non-zero determinant.

## Conclusion

- A **non-singular matrix** is a square matrix with \(\det A \neq 0\), and it is always **invertible**.  
- Its opposite, the **singular matrix**, has \(\det A = 0\) and no inverse.  
- Test by computing the determinant (or checking that rank = n).
- Products, transposes, and non-zero scalar multiples of non-singular matrices stay non-singular.