In Exercises 49-56, use an inverse matrix to solve (if...

In Exercises $49-56,$ use an inverse matrix to solve (if possible) the system of linear equations.
$$
\left\{\begin{array}{c}{-\frac{1}{4} x+\frac{3}{8} y=-2} \\ {\frac{3}{2} x+\frac{3}{4} y=-12}\end{array}\right.
$$

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Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all the equations simultaneously. This concept is foundational in linear algebra and is used to model a wide range of problems in various fields.

Matrix Representation of Linear Systems

A system of linear equations can be represented in matrix form as AX = B, where A is the coefficient matrix containing the coefficients of the variables, X is the column vector of variables, and B is the constant column vector. This representation is essential because it allows the use of matrix operations to solve the system efficiently.

Inverse Matrix

The inverse of a square matrix A is another matrix, denoted A?¹, such that when it is multiplied by A, the result is the identity matrix. In the context of solving linear systems, if the inverse of the coefficient matrix exists, it provides the solution by computing X = A?¹B.

Determinant

The determinant is a scalar value computed from a square matrix and provides important information about the matrix, such as whether an inverse exists. A matrix has an inverse if and only if its determinant is nonzero. This concept is crucial when using matrix inversion methods to solve linear equations.

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