Solve each system by using the inverse of the coefficient matrix. .2 x+.3 y=-1.9 .7 x-.2 y

step 1 The topic of this question is matrix inverses. We're going to be using an inverse matrix to find

Solve each system by using the inverse of the coefficient...

Key Concepts

Inverse Matrix Method

This is the process of solving a system of linear equations by computing the inverse of the coefficient matrix and then using it to solve for the variable vector. Specifically, if the system is represented in the form Ax = b, then the solution can be obtained as x = A?¹b, provided that A?¹ exists. This method streamlines solving systems, especially for systems with multiple equations and variables.

Matrix Inverse

The matrix inverse of a square matrix is a matrix that, when multiplied with the original matrix, yields the identity matrix. It is a crucial concept because the existence of an inverse (which occurs only if the determinant of the matrix is non-zero) allows one to solve linear systems succinctly using algebraic manipulation.

Coefficient Matrix

The coefficient matrix is formed by taking the coefficients of the variables from each equation in the system and arranging them in a matrix. This matrix compactly represents the linear structure of the system, and it is central to methods that involve matrix operations, such as finding the inverse to solve the system.

System of Linear Equations

This concept refers to a collection of two or more equations with the same set of variables. In a system, a solution is a set of values for the variables that satisfies all equations simultaneously. Understanding systems of linear equations is fundamental in linear algebra, as they frequently arise in modeling real-world problems and can be solved using various algebraic techniques.

Transcript

00:01 The topic of this question is matrix inverses.

00:04 We're going to be using an inverse matrix to find the solution to this system of equations.

00:12 To start off, we want to turn this system of equations into an equation involving the coefficient matrix, the unknown vector, and the right side vector.

00:28 To do this, simply take away the x and y's to get the coefficient matrix, and then add in the unknown vector, with x and y in it.

00:54 That will be our left side, and the right side is just the vector with these two numbers in it.

01:11 Now we want to solve for x and y, and we want to do that by finding the inverse of the coefficient matrix.

01:23 So let's find the inverse matrix of this matrix using this method of the augmented matrix.

01:31 So we want to start with this augmented matrix with the left matrix being the one we want to find the inverse four, and the right being an identity matrix.

01:44 Then we do row transformations until the left side becomes the identity matrix, at which point the right side will become our inverse matrix.

01:57 First, let's multiply both rows by 10, just to be things cleaner.

02:03 Now let's add twice the first row to three times the second row.

02:09 So 2 times 2 plus 3 times 7 will be 25.

02:17 2 times 3 plus 3 times negative 2 is 0.

02:21 That's why we want to have this type of combination to cancel out to the entry in the second row...

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