When is an upper triangular matrix invertible

Therefore, in a lower triangular matrix all the elements above the main diagonal i. Definition A matrix is upper triangular if and only if whenever. Thus, in an upper triangular matrix all the elements below the main diagonal i.

Example Consider the matrix The entries on the main diagonal are The entries above the main diagonal are all zero: Therefore, the matrix is lower triangular. Example Define the matrix The entries on the main diagonal are The entries below the main diagonal are all zero: Therefore, the matrix is upper triangular.

Proposition The transpose of a lower triangular matrix is upper triangular. Suppose that is lower triangular, so that whenever. By definition, the entries of the transpose satisfy Therefore, whenever.

Hence, is upper triangular. Proposition The transpose of an upper triangular matrix is lower triangular. Proposition The product of two lower triangular matrices is lower triangular.

Suppose that and are two lower triangular matrices. We need to prove that whenever. But, when , we have that where: in step we have used the fact that because ; in step we have used the fact that because.

Proposition The product of two upper triangular matrices is upper triangular. Proposition A triangular matrix upper or lower is invertible if and only if all the entries on its main diagonal are non-zero. Let us first prove the "only if" part. Suppose a lower triangular matrix has a zero entry on the main diagonal on row , that is, Consider the sub-matrix formed by the first rows of. The -th column of is zero because , and all the columns to its right are zero because is lower triangular.

Then, has at most non-zero columns. As a consequence, it has at most linearly independent columns. ST is the new administrator.

Linear Algebra Problems by Topics The list of linear algebra problems is available here. Subscribe to Blog via Email Enter your email address to subscribe to this blog and receive notifications of new posts by email.

Sponsored Links. Search for:. MathJax Mathematical equations are created by MathJax. Connect and share knowledge within a single location that is structured and easy to search. Is this correct? I think it's still too complicated and perhaps wrong. An upper triangular matrix is invertible if and only if it has no zeros on the main diagonal. Here are some ways to see this:. The determinant of such a matrix is the product of the diagonal entries, and is non-zero if and only if the condition above holds.

The inverse of the matrix can be explicitly computed via row operations. Use the bottom row to clean out the last column, the second to bottom row to clean out the second to last column, and so on. Do you know the criterion for invertibility in terms of the determinant? Sign up to join this community. The product of upper triangular matrices is an upper triangular matrix. The sum or difference of lower triangular matrices is a lower triangular matrix.

The sum or difference of upper triangular matrices is an upper triangular matrix. A triangular matrix is invertible has an inverse if and only if none of its entries in the main diagonal is zero. The inverse of an invertible upper triangular matrix is an upper triangular matrix. The inverse of an invertible lower triangular matrix is a lower triangular matrix. Examples with Solutions Example 1 Which of the following matrices is an upper triangular matrix, a lower triangular matrix or none?