Prove that if A is upper triangular and B is the matrix

Thread starter DavidLiew
Start date Apr 11, 2010
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Matrix

In summary, an upper triangular matrix is a square matrix with zeros below the main diagonal and nonzero elements on or above the diagonal. In the proof, A being upper triangular allows for the use of the property that multiplying two upper triangular matrices results in another upper triangular matrix. B being a matrix is important because the proof involves matrix multiplication. An example is provided to illustrate the proof, and this proof has applications in real-world situations such as engineering and data analysis.

Apr 11, 2010

DavidLiew

How to prove that if A is upper triangular and B is the matrix that results when the ith row and jth column of A are deleted, then B is upper triangular if i<j.

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Apr 11, 2010

Bacle

It seems like induction could work here. Have you tried it.? . Try showing it

works for small values of n, say , 2,3,4 , and see how to generalize for the

inductive step.

1. What is an upper triangular matrix?

An upper triangular matrix is a square matrix in which all the elements below the main diagonal (from top left to bottom right) are zeros. The elements on or above the main diagonal may be nonzero.

2. What is the significance of A being upper triangular in this proof?

In this proof, A being upper triangular means that the elements below the main diagonal are all zeros. This allows us to use the property of upper triangular matrices, namely that multiplying two upper triangular matrices results in another upper triangular matrix.

3. Why is it important that B is a matrix in this proof?

It is important that B is a matrix because we are proving a property that involves matrix multiplication. Without B being a matrix, we cannot use the properties and rules of matrix multiplication to prove the statement.

4. Can you provide an example to illustrate this proof?

Yes, for example, let A be the following 3x3 upper triangular matrix:

A = [2 1 0; 0 4 3; 0 0 5]

And let B be the following 3x3 matrix:

B = [1 2 3; 4 5 6; 7 8 9]

Then, AB is also an upper triangular matrix:

AB = [2 5 9; 16 20 24; 35 40 45]

Notice how all the elements below the main diagonal in AB are zeros, proving the property that if A is upper triangular and B is a matrix, then AB is also upper triangular.

5. How does this proof relate to real-world applications?

This proof is relevant in many real-world applications, such as engineering, physics, and computer science. Upper triangular matrices are used to represent systems of linear equations, and the property being proved helps to simplify and solve these systems more efficiently. This proof also has applications in computer graphics and data analysis, where matrix operations are frequently used.