EBK NUMERICAL METHODS FOR ENGINEERS
EBK NUMERICAL METHODS FOR ENGINEERS
7th Edition
ISBN: 8220100254147
Author: Chapra
Publisher: MCG
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Chapter 10, Problem 1P

Use the rules of matrix multiplication to prove that Eqs. (10.7) and (10.8) follow from Eq. (10.6).

Expert Solution & Answer
Check Mark
To determine

To prove: The equations [L][U]=[A] and [L]{D}={B} follow the equation [L]{[U]{X}{D}}=[A]{X}{B} by the use of matrix multiplication.

Explanation of Solution

Given:

The equations, [L][U]=[A] and [L]{D}={B}.

Proof:

Consider a linear equation represented in the matrix form as,

[A]{X}={B} …… (1)

where, is A is a regular matrix of size 3×3, X is a variable matrix of size 3×1 and B is the resultant matrix of size 3×1, equation (1) can be rewritten as,

[A]{X}{B}=0 …… (2)

Suppose equation (2) could be expressed as upper triangular system,

[u11u12u130u22u2300u33]{x1x2x3}={d1d2d3}       …… (3)

Equation (3) could be expressed in matrix notation and rearranged as,

[U]{X}{D}=0    …… (4)

Now assume there is lower diagonal matrix with 1's on the diagonal,

[L]=[100l2110l31l321]

which has property that when equation (4) pre-multiplied by it, equation (2) is the result. That is,

[L]{[U]{X}{D}}=[A]{X}{B} .….. (5)

Use distributive law of matrix multiplication A[B+C]=AB+AC for matrix A, B, C,

[L]{[U]{X}}[L]{D}=[A]{X}{B} …… (6)

Use associative law of matrix multiplication A[BC]=[AB]C in equation (6) as shown below,

{[L][U]}{X}[L]{D}=[A]{X}{B} …… (7)

Comparing LHS and RHS of equation (7), this gives,

[L][U]=[A] and [L]{D}={B}

Hence, proved that the equations [L][U]=[A] and [L]{D}={B} follow the equation [L]{[U]{X}{D}}=[A]{X}{B}.

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EBK NUMERICAL METHODS FOR ENGINEERS

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