Determinant=0 and invertibility

  • Thread starter Jin314159
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In summary, the determinant of a matrix measures how the volume of the unit box changes. A determinant of zero means that the unit box gets squished into smaller dimensions, making it impossible to undo the operations and find an inverse matrix. This is because infinitely many points get sent to the same place, making it undefined. Additionally, the determinant of a product of two matrices is equal to the product of their determinants, so if the product is the identity matrix, both determinants must be non-zero. Thank you to the contributors for providing both geometric and algebraic explanations.
  • #1
Jin314159
Can someone provide an intuitive understanding of why a matrix is not invertible when it's determinant is zero?
 
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  • #2
The determinant measures how the volume of the unit box changes. Unit box here means all the points

{(a,b,c...,d) | 0<= a,b, ..d <=1



Determinant zero means that it gets squished into smaller dimenisions:

eg, for 2x2, the unit square gets sent to a line segment, in 3x3 the unit cube gets sent to either a 2-d or 1-d figure

you can't undo these operations, because infinitely many points get sent to the same place.

eg

|1 0|
|0 0|

sends all the points with the same y coordinate to the same place, and it squashes the unit square to the unit interval.

Is that ok? That's the geometry, we can talk algebraic reasons too.
 
  • #3
A very good "intuitive reason" is that det(AB)= det(A)det(B).

If AB= I then det(A)det(B)= 1 not 0 so neither det(A) nor det(B) can be 0.
 
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Likes MathewsMD
  • #4
Thanks guys for both the geometric and algebraic intuition.
 
  • #5
To find a inverse matrix, you must take 1/det. If your det is equal to zero, it is undifined.

Paden Roder
 

Determinant = 0 and Invertibility

Understanding the concept of the determinant of a matrix and its relationship to the invertibility of the matrix is crucial in linear algebra and mathematics. Here are some frequently asked questions regarding when the determinant of a matrix is equal to 0 and its implications on invertibility:

Q1: What Is the Determinant of a Matrix?

The determinant of a square matrix is a scalar value that can be calculated from its entries. It is denoted by "det(A)" for a matrix "A." The determinant provides important information about the matrix, including its invertibility and the nature of its solutions in linear systems.

Q2: When Is the Determinant of a Matrix Equal to 0?

The determinant of a square matrix is equal to 0 when the matrix is singular, meaning it does not have an inverse. In mathematical notation, if det(A) = 0 for matrix "A," it implies that "A" is singular and not invertible.

Q3: What Does It Mean for a Matrix to Be Invertible?

A matrix is invertible, or non-singular, if it has an inverse. The inverse of a matrix "A" is denoted as "A^(-1)." If "A" is invertible, then "A^(-1)" exists, and the product of "A" and "A^(-1)" equals the identity matrix, "AA^(-1) = I." Invertible matrices have unique solutions in linear systems and are crucial in various mathematical and engineering applications.

Q4: How Is Determinant = 0 Related to Invertibility?

If the determinant of a matrix is equal to 0 (det(A) = 0), it indicates that the matrix is singular and does not have an inverse. In other words, a matrix with a determinant of 0 is not invertible. This is a fundamental property in linear algebra.

Q5: What Happens in Linear Systems When the Matrix Is Not Invertible (Determinant = 0)?

When the matrix in a linear system is not invertible (determinant = 0), the system may have infinitely many solutions or no unique solution at all. In such cases, the system is said to be dependent or underdetermined, and its solutions form a subspace rather than a unique point.

Q6: Can a Matrix with a Determinant of 0 Still Be Used in Mathematical Calculations?

Yes, a matrix with a determinant of 0 can still be used in mathematical calculations and applications, but it has specific properties and limitations. It is important to recognize when a matrix is singular to appropriately address its behavior in linear systems and computations.

Q7: How Can I Calculate the Determinant of a Matrix?

The method for calculating the determinant of a matrix depends on its size and structure. For 2x2 and 3x3 matrices, there are straightforward formulas. For larger matrices, methods like cofactor expansion or row reduction can be used. Determinant calculation is a fundamental topic in linear algebra, and various resources and software tools are available for assistance.

In summary, the determinant of a matrix being equal to 0 signifies that the matrix is singular and not invertible. Understanding this relationship is crucial in linear algebra, where invertible matrices play a key role in solving linear systems and mathematical applications.

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