An m n upper triangular matrix is one whose entries below the main diagonal are 0’s as in Exercise 8 When is a square upper triangular matrix invertible Justify your answer

A square upper triangular matrix is invertible if and only if all its diagonal entries are non-zero.

To see why, suppose we have a square upper triangular matrix A with diagonal entries a11, a22, ..., ann. We can find its inverse by using row operations to transform A into the identity matrix while applying the same operations to an initially empty matrix B. If A is invertible, then this process will eventually result in A being transformed into the identity matrix I, and B will be transformed into A^-1.

However, if any of the diagonal entries of A are zero, then we cannot perform row operations to turn that entry into a non-zero value. This means that we cannot transform A into the identity matrix, and so A is not invertible.

On the other hand, if all the diagonal entries of A are non-zero, then we can perform row operations to turn each diagonal entry into a 1. Specifically, for each diagonal entry aii, we can divide the ith row of A by aii to make that entry equal to 1, and then subtract suitable multiples of that row from all the rows below it to make all the entries below aii equal to 0. This will result in A being transformed into I, and B being transformed into A^-1, showing that A is invertible.

Therefore, a square upper triangular matrix is invertible if and only if all its diagonal entries are non-zero