Math Problem Statement

Solve the given system of equations by using the inverse of the coefficient matrix. Use a calculator to perform the necessary matrix operations.

x 1x1

plus+

4 x 24x2

minus−

4 x 34x3

minus−

x 4x4

equals=

99

7 x 17x1

plus+

x 2x2

plus+

x 3x3

plus+

5 x 45x4

equals=

88

3 x 13x1

minus−

x 2x2

plus+

x 3x3

minus−

x 4x4

equals=

1010

x 1x1

minus−

x 2x2

minus−

7 x 37x3

minus−

7 x 47x4

equals=

3232

Question content area bottom

Part 1

The solution is

x 1x1equals=enter your response here​,

x 2x2equals=enter your response here​,

x 3x3equals=enter your response here​,

and

x 4x4equals=enter your response here.

​(Type integers or simplified​ fractions.)

Solution

To solve this system of linear equations using the inverse of the coefficient matrix, let’s set up the equations in matrix form.

Given equations:

x1+4x24x3x4=99x_1 + 4x_2 - 4x_3 - x_4 = 99 7x1+x2+x3+5x4=887x_1 + x_2 + x_3 + 5x_4 = 88 3x1x2+x3x4=103x_1 - x_2 + x_3 - x_4 = 10 x1x27x37x4=32x_1 - x_2 - 7x_3 - 7x_4 = 32

This can be represented in matrix form as:

AX=BAX = B

where:

A=[1441711531111177],X=[x1x2x3x4],B=[99881032]A = \begin{bmatrix} 1 & 4 & -4 & -1 \\ 7 & 1 & 1 & 5 \\ 3 & -1 & 1 & -1 \\ 1 & -1 & -7 & -7 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}, \quad B = \begin{bmatrix} 99 \\ 88 \\ 10 \\ 32 \end{bmatrix}

To find XX, we calculate X=A1BX = A^{-1}B, where A1A^{-1} is the inverse of matrix AA.

I'll calculate this to find the values for x1x_1, x2x_2, x3x_3, and x4x_4.The solution to the system of equations is:

x1=10.8125,x2=16.4375,x3=5.6875,x4=0.3125x_1 = 10.8125, \quad x_2 = 16.4375, \quad x_3 = -5.6875, \quad x_4 = 0.3125

If you need any further details on the calculation steps, feel free to ask!

Additional Questions to Explore:

  1. What is the determinant of the coefficient matrix, and why is it significant here?
  2. How does matrix inversion work, and under what conditions does an inverse matrix exist?
  3. Could this system be solved using another method, like Gaussian elimination? How would the process differ?
  4. How can matrix operations help in solving larger systems of equations?
  5. What are some real-world applications where solving systems of equations by matrix inversion is useful?

Tip:

When working with decimals, consider rounding or using fractions to maintain precision, especially if results are required in a simplified form.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Linear Equations
Matrix Operations

Formulas

Matrix form: AX = B
Inverse of a matrix: A^-1
Matrix multiplication to solve AX = B: X = A^-1 * B

Theorems

Properties of Matrix Inversion
Solving Linear Systems Using Matrix Inverses

Suitable Grade Level

Grades 10-12