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Use the rules of matrix multiplication to prove that Eqs. (10.7) and (10.8) follow from Eq. (10.6).
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To prove: The equations
Explanation of Solution
Given:
The equations,
Proof:
Consider a linear equation represented in the matrix form as,
where, is A is a regular matrix of size
Suppose equation (2) could be expressed as upper triangular system,
Equation (3) could be expressed in matrix notation and rearranged as,
Now assume there is lower diagonal matrix with
which has property that when equation (4) pre-multiplied by it, equation (2) is the result. That is,
Use distributive law of matrix multiplication
Use associative law of matrix multiplication
Comparing LHS and RHS of equation (7), this gives,
Hence, proved that the equations
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Chapter 10 Solutions
Numerical Methods for Engineers
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