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

Question

Write the augmented matrix for each system and give its dimension. Do not solve. 2x + y + z - 3 = 0 3x - 4y + 2z + 7 = 0 x + y + z - 2 = 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 have a system of three equations where the first equation is five X plus two, Y plus four, Z minus seven is equal to zero. The second equation is six X minus eight, Y plus 11, Z plus 13 is equal to zero. And the third equation is nine X plus two Y plus six Z minus is equal to zero. Here we have four answer choice options, answer choices A through D, each of them being a different variation of an augmented matrix each with three rows and four columns. So here to begin this problem, we first need to recall augmented matrices which have two large brackets along with a vertical line that represents our equal sign in our equations. And then to the left of this vertical line, we will have three columns since we are given a system with three variables. So in alphabetical order, we start with the mut most column which will hold the X coefficients. Then the second column holds the Y coefficients. And the third column holds the Z coefficients. And now to the right of the vertical line, we have just one column which holds the values of the constants. So now taking a look back at our given system of equations, we can notice that all of the constants for each equation are to the left of the equal sign. However, looking at our matrix set up, we know that these constants need to be to the right of the equal sign. So our first step is to just rearrange each of our equations here so that the constants are on the right side. So doing this, our first equation becomes five X plus two, Y plus four Z is equal to seven. The second equation is six X minus eight, Y plus 11 Z is equal to negative 13. And the third equation is nine X plus two, Y plus six Z is equal to 12. So now that we have rearranged each of our given equations, we can begin filling in our augmented matrix starting with the first row, we need to look at our first equation to get our values. And starting in the first column, we see we need our X coefficient, which in our first equation is five, moving to the right to the second column, we have our Y coefficient which is two. And then moving to the third column, we have these D coefficient which is four. And lastly moving to the right of the vertical line, we have the value of our constant which is seven. So now we just repeat this process moving on to the second row where we now look at our second equation for the values starting again with the X coefficient, we have six. For the Y coefficient, we have negative eight for the Z coefficient, we have 11. And for the value of our constant, we have negative 13. And now moving on to our third and final row, we look at now the third given equation and we have the values 926 and 12. So now we have finished filling in our matrix according to our given system of equations. And here we see that we have three different rows. So we have three rows and we also have four columns. So this tells us that we have a three by four matrix. Therefore, our di dimensions are three by four. And so with these values for our matrix, we are left here with answer choice A where again, the values in the first row are and seven. The values in the second row are six, negative 8 11 and negative 13. And the values in the third row are and 12. 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. 2x + y + z - 3 = 0 3x - 4y + 2z + 7 = 0 x + y + z - 2 = 0

Key Concepts

Augmented Matrix

Dimension of a Matrix

Linear Equations

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