Prove that if U is a nonsingular upper triangular matrix, then the diagonal entries of U^-1 are

Prove that if U is a nonsingular upper triangular matrix,...

Key Concepts

Reciprocal Relationship of Diagonal Elements

For an invertible upper triangular matrix, the simplicity of matrix multiplication directly shows that the diagonal entries of its inverse must be the reciprocals of the original matrix's diagonal entries. This occurs because when two upper triangular matrices are multiplied, the diagonal entry of the product is the product of the corresponding diagonal entries. For the inverse, the requirement that the product equals the identity matrix forces a reciprocal relationship on these elements.

Matrix Inversion

Matrix inversion is the process of finding a matrix, which when multiplied with the given matrix, results in the identity matrix. For triangular matrices, the inversion process benefits from the triangular structure, allowing the inverse to be computed in a step-by-step manner where the diagonal and off-diagonal entries are determined systematically.

Upper Triangular Matrices

Upper triangular matrices are matrices in which all entries below the main diagonal are zero. This structure simplifies many computations, including matrix multiplication and inversion, because operations mainly involve the diagonal and elements above it. Their predictable structure is often exploited to derive properties about the matrix as a whole, such as determinant and invertibility conditions.

Nonsingular Matrices

A nonsingular matrix is one that is invertible, meaning there exists another matrix that multiplied by the original yields the identity matrix. In the context of triangular matrices, nonsingularity implies that all diagonal entries are nonzero, which is critical since the determinant (the product of the diagonal entries) must be nonzero for the matrix to have an inverse.

Transcript

00:01 Okay, so for this exercise we're going to suppose that a is going to be a triangular matrix.

00:06 We need to show the following.

00:08 So first we need to show that a is an invertible matrix if and only if each element in diagonal is different from zero.

00:16 And the second thing that we need to show is that the diagonal elements of the inverse matrix if exist, of course, are going to be the reciprocals of the diagonal elements in the matrix a.

00:28 Okay, so let's start with the first one.

00:33 It is not that hard to prove it because basically we know that a matrix a is invertible if and only if the determinant of the matrix is different from zero.

00:51 And we know that the determinant of a triangular matrix, in this case a, corresponds to the multiplication of the diagonal element.

01:02 A1 times a22 times a33 up to a and n.

01:12 So here we have that the determinant of a because it's a triangular matrix is going to be equals to the multiplication of the diagonal elements.

01:21 And here is clear that this is different from zero if each aith -ith is different from zero.

01:33 And with that we end the proof of this first part.

01:39 Okay, as i mentioned, is not that hard.

01:43 The second one, we need to consider that let's suppose that indeed, a is invertible, then we know that the inverse matrix is going to be equals to the adjoin matrix of a divided the determinant of a...

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