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Chapter 6: Problem 52

Describe what is meant by the augmented matrix of a system of linear equations.

Short Answer

Expert verified
An augmented matrix is a matrix formed by appending the columns of constants on the right side of the equal symbol, to the coefficient matrix of a system of linear equations. The two are separated by a vertical bar.

Step by step solution

01

Describing Basic Concept of Matrices

A matrix is essentially an array of numbers written within square brackets. When dealing with systems of linear equations, each row in the matrix represents an equation, and each column represents a coefficient of a variable.
02

Describing the Coefficient Matrix

In a system of linear equations, the coefficient matrix specifically contains the coefficients of the variables in the system. For instance, in the equations \(2x + 3y = 6\) and \(4x - y = 8\), the coefficient matrix would be \(\begin{bmatrix} 2 & 3 \ 4 & -1 \end{bmatrix}\) .
03

Describing the column of constants on the right

We also have the constants on the right side of the equations, for the same system of equations, the right-side constants are [6,8].
04

Describing Augmented Matrix

The augmented matrix combines the coefficient matrix and the column of constants into a single matrix. By using a vertical bar to separate the coefficients from the constants, we arrive at the augmented matrix of the system. For the previous system of equations it will be: \(\begin{bmatrix} 2 & 3 | & 6 \ 4 & -1 | & 8 \end{bmatrix}\) .

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Most popular questions from this chapter

Use a graphing utility to evaluate the determinant for the given matrix. $$\left[\begin{array}{rrrr}3 & -2 & -1 & 4 \\\\-5 & 1 & 2 & 7 \\\2 & 4 & 5 & 0 \\\\-1 & 3 & -6 & 5\end{array}\right]$$

If \(I\) is the multiplicative identity matrix of onder \(2,\) find \((I-A)^{-1}\) for the given matrix \(A\) $$\left[\begin{array}{rr}8 & -5 \\\\-3 & 2\end{array}\right]$$

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) (In Exercises \(59-60\), be surre to refer to matrix \(B\) described in the second column on the previous page.) a. If \(A=\left[\begin{array}{ll}2 & 0 \\ 0 & 1\end{array}\right],\) find \(A B\) b. Graph the object represented by matrix \(A B\). What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(\vec{B}\) ?

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's Rule.

Describe how to perform scalar multiplication. Provide an example with your description.
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