In Problems 41-60, use the inverses found in Problems 31-40 to solve each system of equations. {

In Problems 41-60, use the inverses found in Problems 31-40...

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Key Concepts

Solving Linear Systems Using Inverse

This method involves finding the inverse of the coefficient matrix to directly solve the system of equations. Once A?¹ is computed, multiplying it by the constant vector b yields the solution vector x, providing a systematic approach to solving the system when the inverse exists.

Matrix Inverse

The matrix inverse is a key concept when solving systems using the equation Ax = b. If A is invertible, then the solution can be obtained by computing x = A?¹b. The existence of an inverse requires that A be a square matrix with non-zero determinant.

System of Linear Equations

This concept involves equations that contain multiple unknowns where each equation represents a linear relationship between those unknowns. The overall system is solved by finding a common set of values that satisfy all the equations simultaneously.

Matrix Representation

A system of linear equations can be written in matrix form as Ax = b, where A is the coefficient matrix describing the relationships between variables, x is the column vector of the variables, and b is the column vector of constants. This compact representation facilitates the application of linear algebra techniques.

Transcript

00:01 Alright, we're going to solve the following uses inverses, so writing out our different matrixes.

00:06 Our a matrix is 1 -1 -1, 3 -2 -negative 1, 312.

00:16 Our x -matrix are our variables, so x, y, z.

00:22 Our b matrix is our solution, so 9 -8 -1.

00:29 And our inverse we found in the previous section, so we just have to make.

00:32 Match it up.

00:33 I've got them all written on the next page.

00:35 So we're looking for the one that matches to 1 -1 -1 -1.

00:38 So it's negative 5 over 7, 1 over 7, 3 over 7.

00:43 Negative 5 over 7, 1 over 7, 3 over 7.

00:53 9 over 7, 1 over 7, negative 4 over 7.

01:02 And then 3 over 7, negative 2 over 7, and 1 over 7.

01:10 Over seven, negative two over seven, one over seven.

01:14 So we're just gonna double check that again.

01:19 9 -1, negative four, three, negative two, one.

01:23 Three, negative, two, one.

01:25 Perfect.

01:26 All right, so once we have that, we're just gonna take our inverse.

01:31 Five over seven, nine over seven, three over seven, one over seven, one over seven, one over seven, negative two over seven, 3 over 7, negative 4 over 7, 1 over 7...

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