Good evening Could you help me in this equestion please Show

Good evening

Could you help me in this equestion, please?

Show that if a triangular square matrix is nonsingular, its diagonal entries are all nonzero.

Thank you in advance

Solution

A square matrix, A, is non-singular if and only if its determinant, det (A), is non-zero.

A determinant can be found by choosing any row (or column), multiplying each entry in that row by its minor, and summing.

For a triangular nxn matrix, suppose the triangle of 0\'s is above the diagonal. Then choose the top row. All elements in it but the first are 0, so the determinant is that first entry times the determinant of the (n-1)x(n-1) left over after crossing out the first row and first column.

Now do it again to find that determinant. You have a (n-1)x(n-1) matrix whose first row is 0 except for its first element.

Continuing in this way, it becomes obvious that det (A) = product of the diagonal elements. This product is nonzero, and therefore A is non-singular, if and only if all of those elements are nonzero.