Matrix Invertability & Singularity: Explained

In summary, the conversation discussed whether a matrix can be both invertible and singular. The answer is no, as being invertible means it is non-singular. The link provided explains the concept of pseudo-inverse, which is not the same as inverse. It was also clarified that a singular matrix has a zero determinant and its inverse cannot be obtained, while a non-singular matrix has a non-zero determinant and can be inverted.
  • #1
phymatter
131
0
if a matrix is invertable can it be singular ?
 
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  • #3
The link says it calculates the pseudo-inverse. The pseudo-inverse is a well known and useful concept, but it is NOT the same as the inverse.

The answer to the OP's questiion is "no".
 
  • #4
No, if a matrix is invertible it is said to be non-singular which is the exact opposite of singular.
In other words a singular matrix has got a zero determinant and as such it's inverse cannot be obtained.A non singular matrix has got the determinant not equal to zero and in the due course it's inverse can be obtained hence the name invertible matrix.
 
  • #5
@AlephZero and kaliro - Point conceeded.

@phymatter please accept my apologies if my post was misleading.
 
  • #6
thanks for everyone's help :)
 

1. What is matrix invertibility?

Matrix invertibility is the property of a square matrix where it can be multiplied by another matrix to produce the identity matrix, which is a square matrix with 1s on the main diagonal and 0s everywhere else. In simpler terms, a matrix is invertible if it has an inverse that can "undo" its operations.

2. Why is matrix invertibility important?

Matrix invertibility is important because it allows us to solve linear systems of equations and perform other mathematical operations such as finding determinants and eigenvalues. It also has applications in various fields such as physics, engineering, and computer graphics.

3. What does it mean for a matrix to be singular?

A matrix is singular if it is not invertible, meaning it does not have an inverse. This happens when the determinant of the matrix is equal to 0. Geometrically, a singular matrix represents a transformation that collapses a space onto a lower-dimensional subspace.

4. How do you determine if a matrix is invertible?

To determine if a matrix is invertible, you can calculate its determinant. If the determinant is not equal to 0, then the matrix is invertible. Alternatively, you can also check if the matrix has full rank, which means that all of its columns or rows are linearly independent.

5. Can a non-square matrix be invertible?

No, a non-square matrix cannot be invertible. Only square matrices have inverses, and the dimensions of the inverse matrix will be the same as the original matrix. However, there are some special types of non-square matrices, such as the pseudoinverse, that can be used in certain cases to "invert" a non-square matrix.

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