Demonstrate that a matrix that has a null row is not invertible

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In summary, to demonstrate that a matrix with a null row is not invertible, we can show that it cannot be equivalent to the identity matrix. This is because the product of the matrix and its inverse should result in the identity matrix, but a null row would result in a null column, proving that the matrix does not have an inverse. This is a valid proof, but its suitability may depend on the expectations of the teacher.
  • #1
fluidistic
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Homework Statement


Demonstrate that a matrix that has a null row is not invertible.

2. The attempt at a solution I know that saying that a matrix A is invertible is equivalent to say that A is row-equivalent and column-equivalent to the identity matrix.
And also that there exist elemental matrices E_1, E_2,...,E_r and F_1,...,F_s such that the identity matrix is equal to E_1...E_r A F_1...F_s.
So I could show that if A has at least a null row, then one of these properties cannot be true.
But still I don't know how to proceed. I understand clearly that if A has a null row, it's obvious that it cannot be equivalent to the identity matrix... but to prove it formally, I don't know how I can start. A little help is welcome.
 
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  • #2
Suppose it did have an inverse. What does its product with its inverse supposed to look like? How does it actually look?
 
  • #3
Think of A as the coefficient matrix such that Ax= b and let xn be the component of x corresponding to the 0 row. If the corresponding component of b is not 0, there is NO value of xn that makes that true. If the corresponding component of b is 0, then any value of xn makes that true. In either case there is NOT single solution to Ax= b. If A-1 exists, then A-1b would be 'the' solution to Ax= b.
 
  • #4
Hurkyl said:
Suppose it did have an inverse. What does its product with its inverse supposed to look like? How does it actually look?

It is supposed to look like the identity matrix, however if the matrix has a null row it means that the product of this matrix with its inverse has a null column which means it's not the identity matrix, so that in fact any matrix having a null row doesn't have an inverse, hence is not invertible.
Is it a good proof? Or too informal?
 
  • #5
fluidistic said:
Is it a good proof? Or too informal?
I can't answer that one; it would depend on what your teacher expects.
 
  • #6
Hurkyl said:
I can't answer that one; it would depend on what your teacher expects.
Ok I understand. I hope it's enough.
And thank you very much HallsOfIvy, I got what you mean... nice way to do the proof.
 

Related to Demonstrate that a matrix that has a null row is not invertible

1. Why is it important to demonstrate that a matrix with a null row is not invertible?

Demonstrating that a matrix with a null row is not invertible is important because it helps us determine the properties and behavior of matrices. Invertible matrices have unique solutions and can be used to solve systems of linear equations. Understanding when a matrix is not invertible helps us avoid errors and find alternative methods for solving problems.

2. What does it mean for a matrix to have a null row?

A null row in a matrix means that all the elements in that row are equal to zero. This can also be referred to as a row of zeros.

3. How can I tell if a matrix has a null row?

To determine if a matrix has a null row, you can simply scan each row and check if all the elements are equal to zero. Another way is to use Gaussian elimination to reduce the matrix to row-echelon form, where a null row would be represented by a row of zeros.

4. Can a matrix with a null row ever be invertible?

No, a matrix with a null row can never be invertible. In order for a matrix to be invertible, it must have a non-zero determinant. A determinant of zero indicates that the matrix lacks unique solutions and is not invertible.

5. What are some real-world applications that demonstrate the importance of understanding invertible matrices?

Invertible matrices are essential in fields such as engineering, physics, and economics. In engineering, invertible matrices are used to solve systems of equations in structural analysis and fluid dynamics. In physics, they are used in calculations related to electric circuits and quantum mechanics. In economics, invertible matrices are used to model and analyze economic systems.

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