Use Cramer's Rule to solve the system of equations. { -2 x+3 y-z =7 x-2 z =-7 -3 y+z

Use Cramer's Rule to solve the system of equations. { -2 x+3...

Key Concepts

Cramer's Rule

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns. It involves calculating the determinants of matrices formed by replacing one column of the coefficient matrix with the constants from the equations, and then dividing by the determinant of the coefficient matrix to find the value of each variable.

Determinant

A determinant is a scalar value computed from a square matrix, and it encapsulates important properties of the matrix. In the context of solving linear systems using Cramer's Rule, the determinant of the coefficient matrix must be non-zero to ensure that the system has a unique solution.

Coefficient Matrix

The coefficient matrix is a matrix composed entirely of the coefficients of the variables in a system of linear equations. This matrix is central to applying matrix methods like Cramer's Rule, as its determinant is used to determine the existence and uniqueness of the solution to the system.

Systems of Linear Equations

Systems of linear equations consist of multiple linear equations that share the same set of variables. These systems can be solved using various methods, including substitution, elimination, and matrix techniques such as Cramer's Rule. Their study is fundamental in linear algebra and has applications in numerous fields including engineering and economics.

Transcript

00:01 This question gives us a system of equations and asks us to use kramer's rule to find a solution.

00:06 The system that it gives is this.

00:09 Negative 2x plus 3y minus z equals 7.

00:17 X minus 2z equals negative 7 and negative 3y plus z equals negative 1.

00:33 So since we're going to use kramer's rule, we want to first turn this into a matrix equation.

00:37 A matrix equation looks like this.

00:39 First we're going to start with the coefficient matrix.

00:42 That is, we're going to have a column for all the x coefficients, all the y coefficients, all the y coefficients, and the z coefficients.

00:47 So the first equation, there are negative two x.

00:50 In the second one, there's only one x.

00:52 And in the third, there are zero x.

00:55 For the y is it's 3, 0, and negative 3.

01:00 And for the zs, it's negative 1, negative 2, and positive 1.

01:06 Then we will have our variable matrix.

01:10 That's just going to be x, y, and z.

01:12 Those are our variables in that order.

01:14 And then that will be equal to our constant coefficient matrix, 7, negative 7, and negative 1.

01:23 So this is our matrix equation, and now we can use this to start doing kramer's rule.

01:28 Now kramer's rule, we know, this tells us that x is equal to dx divided by d.

01:35 Now what is this? we'll get to dx in a second, but d we already know.

01:41 D is the determinant of this matrix, the coefficient matrix.

01:46 So d is just the determinant of the matrix, negative 2, 3, negative 1, 1, 0, negative 2, and 0, negative 3, 1.

01:57 So when finding that determinant, we're pros at finding determinants of 3x3 matrices by now.

02:04 So i'm not going to show the whole process of co -factor expansion, but the determinant of this matrix is 12.

02:11 So now we know that d is 12.

02:13 All we need to do now is find dx.

02:15 With dx, we're going to do the same thing except change a little bit.

02:19 The x column, remember we had a column for all the x coefficients.

02:23 In that matrix, we're going to replace the x column with our constants.

02:27 So instead of negative 2, 1, 0, we'll have 7, negative 7, and 1...

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