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To introduce triangular matrices and LU-Decomposition

To learn how to use an algorithmic technique in order to decompose arbitrary matrices

To apply LU-Decomposition in the solving of linear systems

This packet introduces triangular matrices, and the technique of decomposing matrices into triangular matrices in order to more easily solve linear systems.

Tutorial

Before beginning with this packet, you should be comfortable with matrix multiplication, Gaussian elimination, the definition of the determinant of a matrix (see also here), and solving linear systems.

The determinant of a triangular matrix, either upper or lower, and of any size, is just __ the product of its diagonal entries__. This single property immensely simplifies the ordinarily laborious calculation of determinants. Here are some examples for 2x2 and 3x3 matrices.

*Lower Triangular 2x2 Matrix*

has a determinant of * ad-c0 = ad*.

*Upper Triangular 3x3 Matrix*

which has a determinant of * a(ej - 0f) - b(0j - 0f) + c(0 - 0e) = aej*.

Here is a numerical example of the same thing.

which has the determinant * 1(2*2 - 0*1) - 2(0*2 - 0*1) + 3(0*0 - 0*2) = 1*2*2 = 4*, which is just the product of the diagonal entries.

LU-Decomposition

Solving Linear Systems

In this example we find an LU Decomposition for a matrix.

Here is a screen capture using the free computer algebra package called maxima to check our work:

In the above, we define * L_{2}* and

We use LU-Decomposition to solve a linear system.