Math Problem Statement
Matrix inverse: Solve each system of equations by using the inverse of the coeffcient matrix if it exists and by the Gauss-Jordan method if the inverse doesn’t exist. 2x + y =1 3y + z =8 4x - y -3z =8
Solution
We are given the system of equations:
Step 1: Write the system as a matrix equation
This system of equations can be expressed in matrix form as:
Where:
\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 1 \\ 8 \\ 8 \end{pmatrix}$$ ### Step 2: Check if the inverse of the matrix $$A$$ exists To determine whether the inverse of $$A$$ exists, we first calculate its determinant. If the determinant is non-zero, the inverse exists. The determinant of a 3x3 matrix $$A$$ is given by: $$\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ For matrix $$A$$: $$A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 1 \\ 4 & -1 & -3 \end{pmatrix}$$ Let me calculate the determinant.The determinant of matrix $$A$$ is $$-12$$, which is non-zero. Therefore, the inverse of $$A$$ exists, and we can solve the system using the inverse of the coefficient matrix. ### Step 3: Solve the system using the inverse of $$A$$ The solution to the system is given by: $$\mathbf{x} = A^{-1} \mathbf{b}$$ Let me calculate the inverse of $$A$$ and use it to find $$\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$.The solution to the system of equations using the inverse of the coefficient matrix is: $$x = -2, \quad y = 5, \quad z = -7$$ ### Summary: - $$x = -2$$ - $$y = 5$$ - $$z = -7$$ Would you like further details on the inverse matrix or Gauss-Jordan method? Here are 5 related questions to deepen your understanding: 1. How do we calculate the inverse of a matrix manually? 2. What is the Gauss-Jordan elimination method? 3. How do we interpret the determinant of a matrix geometrically? 4. Can every matrix be inverted? What are the conditions? 5. What is the relationship between the determinant and matrix singularity? **Tip:** Always check the determinant before attempting to compute the inverse of a matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Inversion
Gauss-Jordan Elimination
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Matrix equation: A * x = b
Solution using inverse matrix: x = A^{-1} * b
Theorems
Inverse Matrix Theorem
Determinant Calculation
Gauss-Jordan Elimination
Suitable Grade Level
Undergraduate (Linear Algebra)
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