# Proving Triangular Matrix Inverse is Also Triangular

• crd
In summary, it is a well-known fact in linear algebra that the inverse of a triangular matrix is also triangular. However, there are no accompanying proofs for this statement. Some have tried to prove it using row operations on the matrix, but this method is not very elegant. Another approach is to use induction, starting from the last row and column of the matrix and working your way up to the upper left corner. This ultimately proves that the inverse of a triangular matrix is indeed triangular.
crd
It is stated in almost every linear algebra text i could find that the inverse of a triangular matrix is also triangular, but no proofs accompanied such statements.

I am convinced that it is the truth, but I have not been able to write anything down that I am satisfied with that doesn't rely on the argument that row operations on the matrix (A|I) to obtain (I|A^{-1}).

Since this would only be the forward pass(if A is lower triangular) and the backwards pass(if A is upper triangular) and these operations ultimately do not introduce non zero terms above/below the diagonal entries(depending on what A was), thus A^{-1} would be a triangular matrix of the same flavor.

Has anyone come across anything a little more elegant than simply brute forcing it?

Assume you have a invertible upper triangular matrix. Consider

$$AA^{-1} = I$$

You can use induction, starting from the last row of A times the last column of $A^{-1}$. gives you the entry. lower left entry 1.

Then again take the last row of A and n-1 column of $A^{-1}$. To be able to get a zero in the identity matrix, (n,n-1) entry of $A^{-1}$ must be zero.
...
Carry on to the upper left corner and you are done.

## 1. What is a triangular matrix?

A triangular matrix is a type of square matrix where all elements above or below the main diagonal are equal to zero. Depending on the position of the main diagonal, a triangular matrix can be an upper triangular matrix (all non-zero elements are on or above the diagonal) or a lower triangular matrix (all non-zero elements are on or below the diagonal).

## 2. Why is proving triangular matrix inverse important?

Proving that a triangular matrix inverse is also triangular is important because it has several applications in linear algebra and other areas of mathematics. Triangular matrices are easier to work with and can simplify complex calculations, so knowing that their inverse is also triangular can make solving problems more efficient.

## 3. How can you prove that a triangular matrix inverse is also triangular?

The most common approach to proving that a triangular matrix inverse is also triangular is by using the Gauss-Jordan elimination method. This method involves performing elementary row operations on the original matrix until it is reduced to the identity matrix. The resulting matrix will be the inverse of the original triangular matrix, and since the inverse of an upper (lower) triangular matrix is also upper (lower) triangular, the proof is complete.

## 4. Can a non-triangular matrix have a triangular inverse?

No, a non-triangular matrix cannot have a triangular inverse. The inverse of a matrix is only defined when the matrix is square and non-singular (determinant is not equal to zero). Additionally, the inverse of a triangular matrix must also be triangular, as proven in the previous question.

## 5. Are there any other methods for proving a triangular matrix inverse is also triangular?

While Gauss-Jordan elimination is the most commonly used method for proving a triangular matrix inverse is also triangular, there are other approaches as well. One method involves using the property that the product of two upper (lower) triangular matrices is also upper (lower) triangular. By using this property and the fact that the inverse of a matrix is equal to the product of its adjugate and the reciprocal of its determinant, it can be proven that the inverse of a triangular matrix is also triangular.

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