Numerical Methods for Engineers
Numerical Methods for Engineers
7th Edition
ISBN: 9780073397924
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Textbook Question
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Chapter 9, Problem 1P

(a) Write the following set of equations in matrix form:

8 = 6 x 3 + 2 x 2 2 x 1 = x 3 5 x 2 + 8 x 1 = 13

(b) Multiply the matrix of coefficients by its transpose; i.e., [ A ] [ A ] T .

(a)

Expert Solution
Check Mark
To determine

The matrix form of the following set of equations,

8=6x3+2x22x1=x35x2+8x1=13

Answer to Problem 1P

Solution:

The matrix form of the set of equations is, [026101850]{x1x2x3}={8213}.

Explanation of Solution

Given:

The set of equations,

8=6x3+2x22x1=x35x2+8x1=13

Consider the set of equations,

8=6x3+2x22x1=x35x2+8x1=13

The set of equations can be written as,

0x1+2x2+6x3=8x1+0x2+x3=28x1+5x2+0x3=13

These equations have three unknowns x1, x2 and x3.

The matrix form of the above set of equations can be written as,

[A]{X}={B}

Where, A is the coefficient matrix.

Therefore, the matrix form of the set of equations is,

[026101850]{x1x2x3}={8213}.

(b)

Expert Solution
Check Mark
To determine

To calculate: The matrix multiplication of the coefficient matrix with its transpose, that is [A].[A]T. Where the coefficient matrix is, [A]=[026101850].

Answer to Problem 1P

Solution:

The matrix multiplication of the coefficient matrix with its transpose is,

[A].[A]T=[4061062810889]

Explanation of Solution

Given:

The coefficient matrix, [A]=[026101850].

Formula used:

Transpose of the matrix A is denoted by AT. If A is any matrix [abcdefghi], then its transpose will be [adgbehcfi].

Calculation:

Consider the coefficient matrix,

[A]=[026101850]

The transpose of the matrix A is denoted by [A]T and is equal to,

[A]T=[018205610]

Multiplication of the coefficient matrix and its transpose is,

[A].[A]T=[026101850].[018205610]=[0×0+2×2+6×60×1+2×0+6×10×8+2×5+6×01×0+0×2+1×61×1+0×0+1×11×8+0×5+1×08×0+5×2+0×68×1+5×0+0×18×8+5×5+0×0]=[4+3661062810864+25]=[4061062810889]

Therefore, the matrix multiplication of the coefficient matrix with its transpose is,

[A].[A]T=[4061062810889].

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2 Q/ Let d₂ +d, di, d2: R² XR² R² defined as follow ((x+x), (2, 1) = √(x-2)² + (x_wx • d₁ ((x,y), (z, w)) = max {1x-z\, \y-w\} • 1 1 dq ((x,y), (Z, W)) = \ x=2\+\-w| 2 • show that dod₁, d₂ are equivalent? 2
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