Use an inverse matrix to solve (if possible) the system of linear equations. { -0.4 x+0.8 y=

step 1 It is given that system of linear equations minus 0 .4 x plus 0 .8y is equal to be 1 .6 and 2x m

Use an inverse matrix to solve (if possible) the system of...

Key Concepts

System of Linear Equations

A system of linear equations consists of multiple linear equations that share the same set of variables. The goal is to find values for these variables that satisfy every equation simultaneously. These systems are fundamental in linear algebra and are widely used in various fields such as engineering, economics, and sciences.

Matrix Representation

Instead of handling each equation separately, 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 column vector of constants. This representation facilitates the use of linear algebra techniques to solve the system.

Determinant

The determinant of a matrix is a scalar value that can be computed from its elements and provides important information about the matrix. In the context of solving systems of linear equations, the determinant of the coefficient matrix indicates whether the matrix is invertible. A non-zero determinant means the matrix is invertible and the system has a unique solution, whereas a zero determinant suggests that the system may have no unique solution.

Inverse Matrix

The inverse of a square matrix A, denoted A?¹, is a matrix such that when it is multiplied by A, it yields the identity matrix. The inverse exists only if the determinant of A is non-zero. Calculating the inverse matrix is a key step in solving systems of linear equations, as it allows one to isolate the variable vector x.

Solution using the Inverse Matrix Method

When the coefficient matrix A of a system of linear equations is invertible, the system can be solved uniquely by multiplying both sides of the equation Ax = b by A?¹. This results in x = A?¹b, effectively using the inverse matrix to determine the values of the variables that satisfy all equations in the system.

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