Write the augmented matrix for each system and give its dimens...

Question

Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y + z - 7 = 0 5x - y + 4z - 7 = 0

Accepted Answer

Hey, everyone. Here we are asked to find the augmented matrix for the given system of equations as well as state the dimensions of the matrix. So here we are given a system of three equations where the first equation is X minus three Y plus five, Z minus six is equal to zero. The second equation is seven X plus two, Y plus 10, Z minus nine is equal to zero. And the third equation is three X minus eight Y plus two, Z minus 13 is equal to zero. Here we have four answer choice options, answer choices A through D, each of them being an augmented matrix with three rows and four columns containing different variations of values. So here to begin this problem, we first need to recall the setup of augmented matrices where we first have two large brackets along with a vertical line that represents our equal sign into the right of this vertical line. We will have one column which will hold the values of our constants into the left of this vertical line. We will have three columns, one for each variable where in alphabetical order, starting in the leftmost column, we will have a column for the X coefficients. Then moving to the second column, we will have our Y coefficients and then the third column, we will have our Z coefficients. So now looking back at our given system of equations, we can notice that our constant values are to the left of the equal sign. Well, according to our matrix here, we need these constant values to the right side of the equal sign. And so our first step is to just rearrange each of our equations. So our first equation is now X minus three Y plus five Z is equal to six. Our second equation is then seven X plus two, Y plus 10, Z is equal to nine. And our third equation is then three X minus eight Y plus two Z is equal to 13. So now that we have rearranged each of our equations, we can begin filling in the values of our matrix. So starting with the first row, we need to look at our first equation to obtain the values. And so beginning in the first column, we need the coefficient of X, which we see is one, moving to the second column. We need the coefficient of Y which is negative three. Now, in the third column, we need the coefficient of Z which is five. And now in our last column, we need the value of the constant, which is six. And now we can move on to our second row where we now look at our second equation for the values. And so the coefficient of X is seven, the coefficient of Y is two, the coefficient of Z is 10 and the value of our constant is nine. And now we repeat this process one last time in our third row where we now look at our third equation for the values and we have the values three negative 82 and 13. So now that we have filled in our matrix, we see that we have three rows, one for each equation. And we also have four columns. So this tells us we have a three by four matrix. So those are, are our dimensions. And with these values, we are left with answer choice D where again, the values in the first row are one negative 35 and six. The values in the second row are 72, 10 and nine. And the values in the third row are three negative and 13. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y + z - 7 = 0 5x - y + 4z - 7 = 0

Key Concepts

Augmented Matrix

Matrix Dimensions

System of Linear Equations

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