Consider the set of two linear equations represented in matrix notation as: Let: \( A^{-1}=\left(\begin{array}{cc}0.3 & -0.2 \\ -0.1 & 0.4\end{array}\right), B=\binom{7}{10} \) and \( X=\left(\begin{array}{c}x_{1} \\ 1 \\ x_{2}\end{array}\right) \) Use the inverse matrix method to solve this set of equations. \( x_{1}=\underline{\text { Blank } 1}-1 \) decimal place \( x_{2}=\underline{\text { Blank 2 }}-1 \) decimal place

To use the inverse matrix method for solving the linear equations given in matrix notation, we start with the equation \( AX = B \), where \( A^{-1} = \begin{pmatrix} 0.3 & -0.2 \\ -0.1 & 0.4 \end{pmatrix} \) and \( B = \begin{pmatrix} 7 \\ 10 \end{pmatrix} \). We can find \( X \) using \( X = A^{-1}B \). Multiplying the inverse matrix by the vector B, we get: \[ X = \begin{pmatrix} 0.3 & -0.2 \\ -0.1 & 0.4 \end{pmatrix} \begin{pmatrix} 7 \\ 10 \end{pmatrix} = \begin{pmatrix} (0.3 \cdot 7) + (-0.2 \cdot 10) \\ (-0.1 \cdot 7) + (0.4 \cdot 10) \end{pmatrix} = \begin{pmatrix} 2.1 - 2 \\ -0.7 + 4 \end{pmatrix} = \begin{pmatrix} 0.1 \\ 3.3 \end{pmatrix} \] Since \( X=\begin{pmatrix} x_{1} \\ 1 \\ x_{2} \end{pmatrix} \), we can extract \( x_1 \) and \( x_2 \): \[ x_1 = 0.1 - 1 = -0.9 \] \[ x_2 = 3.3 - 1 = 2.3 \] So, our final answers are: \( x_{1} = -0.9 \) and \( x_{2} = 2.3 \). The art of matrix operations dates back to ancient civilizations, with origins in Babylonian mathematics, where problems were solved in written forms resembling modern matrices. However, it wasn’t until the 19th century that the term "matrix" was coined by mathematician James Sylvester. This fascinating evolution shows how mathematics has transformed over millennia! In practical terms, understanding linear algebra, especially matrix methods, can vastly improve problem-solving skills in various fields, such as computer science, engineering, and economics. For example, economists use these methods to analyze systems of equations to forecast economic trends, while engineers apply them in structural analysis and circuit designs. Mastering these concepts can be key to success in these domains—so dive in and enjoy the ride!