Brucezhou said:This seems easy but when I tried to do this, the best way I came up with is to list all entries and then do the multiplication work. Is there any better ,clearer and more simple way to do the proof?
An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero. The main diagonal is defined as the line of elements from the top left corner of the matrix to the bottom right corner.
The product of two upper triangular matrices is another upper triangular matrix. This means that the resulting matrix will also have all its elements below the main diagonal equal to zero.
To prove that the product of two upper triangular matrices is upper triangular, you must show that all the elements below the main diagonal are equal to zero. This can be done by using the definition of matrix multiplication and the properties of upper triangular matrices.
Yes, induction can be used to prove that the product of upper triangular matrices is upper triangular. You can start by showing that the statement is true for a 2x2 matrix and then assume it is true for an n x n matrix. By using induction, you can then show that it is also true for an (n+1) x (n+1) matrix.
Yes, there are other methods that can be used to prove the product of upper triangular matrices is upper triangular. Some possibilities include using mathematical induction, the properties of matrix multiplication, and the fact that the product of two upper triangular matrices can be written as a sum of diagonal matrices. You can also use counterexamples to show that the statement is not true in certain cases.