Singular Matrix

Singular Matrix

A singular matrix is a square matrix that does not have an inverse. This occurs when its determinant is zero (det(A) = 0), meaning the matrix has no inverse and represents a system of equations with either no solution or infinitely many solutions.

The following diagrams show how to determine if a 2×2 matrix is singular and if a 3×3 matrix is singular. Scroll down the page for examples and solutions.

 

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Example:

Solution:
Determinant = (3 × 2) – (6 × 1) = 0

The given matrix does not have an inverse. It is a singular matrix.

Characteristics and Properties of a Singular Matrix:

1. Determinant is Zero:
   This is the defining characteristic. If you calculate the determinant of a square matrix and get 0, it’s singular.
   
2. No Inverse Exists (Non-Invertible):
   This is a direct consequence of the determinant being zero. The formula for the inverse of a matrix involves dividing by its determinant. Since division by zero is undefined, a singular matrix cannot have an inverse.
   
3. Square Matrix Only:
   The concept of a determinant is only defined for square matrices (matrices with the same number of rows and columns, e.g., 2x2, 3x3, nxn). Therefore, only square matrices can be singular.
   
4. Linearly Dependent Rows/Columns:
   The rows of a singular matrix are linearly dependent. This means at least one row can be expressed as a linear combination of the other rows (e.g., one row is a multiple of another, or one row is the sum of two other rows).
   Similarly, the columns of a singular matrix are linearly dependent.
   
5. Associated Linear System:
   If a singular matrix A is part of a system of linear equations in the form Ax=b, this system will either have no solution or infinitely many solutions. It will never have a unique solution.

When A Matrix Cannot Be Inverted And The Reasons Why It Cannot Be Inverted?

How to know if a matrix is invertible?
How to know if a matrix is singular?

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What Is A Singular Matrix And How To Tell If A 2x2 Matrix Is Singular?

A singular matrix is one which is non-invertible i.e. there is no multiplicative inverse, B, such that the original matrix A × B = I (Identity matrix)

A matrix is singular if and only if its determinant is zero.

Example: Are the following matrices singular?

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Determine A Value In A 2×2 Matrix To Make The Matrix Singular

A square matrix A is singular if it does not have an inverse matrix.

Matrix A is invertible (non-singular) if det(A) = 0, so A is singular if det(A) = 0

Example: Determine the value of b that makes matrix A singular.

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Singular Matrices Examples

1. Determine whether or not there is a unique solution.
2. For what value of x is A a singular matrix.

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Determine A Value In A 3x3 Matrix To Make The Matrix Singular

Example: Determine the value of a that makes matrix A singular.

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