Problem 13 - Precalculus: Mathematics for Calculus

Question

A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in both row-echelon and reduced row-echelon forms. The system of equations is \(x = -3\) and \(y = 5\).

Step 1: Understand Row-Echelon Form

A matrix is in row-echelon form if all nonzero rows are above any rows of all zeroes (there are no zero rows here), the leading entry of a nonzero row is to the right of the leading entry of the row above it, and all entries in a column below a leading entry are zero.Let's check these conditions for the given matrix:\[ \begin{bmatrix} 1 & 0 & -3 \ 0 & 1 & 5 \end{bmatrix} \]1. Both rows are nonzero and there are no zero rows.2. The first nonzero entry in the first row is 1, and the first nonzero entry in the second row is also 1, which is to the right of the leading entry of the first row.3. There are no entries below the leading entries in each row that need to be zero, as each row ends before any entries in the column could appear below.Therefore, the matrix is in row-echelon form.

Step 2: Determine Reduced Row-Echelon Form

A matrix is in reduced row-echelon form if it meets all the criteria for row-echelon form, and additionally: 1. The leading entry in each nonzero row is 1. 2. Each leading 1 is the only nonzero entry in its column. Let's check these additional conditions: 1. The leading entries in both rows are 1. 2. In the given matrix: - The first column has a 1 as the leading entry in the first row and zero everywhere else. - The second column has a 1 as the leading entry in the second row and zero everywhere else. Thus, except for the entries to the right of the leading positions, all other entries in these columns are zero. Therefore, the matrix is in reduced row-echelon form.

Step 3: Write the System of Equations

To convert the matrix into a system of equations, consider each row as a separate equation, equating it to the entries in the last column:The matrix\[\begin{bmatrix}1 & 0 & -3 \0 & 1 & 5\end{bmatrix}\]corresponds to the system:1. The first row is: \( 1x + 0y = -3 \), simplifying to \( x = -3 \).2. The second row is: \( 0x + 1y = 5 \), simplifying to \( y = 5 \).This gives us the system of equations: \[\begin{cases} x = -3 \ y = 5 \end{cases}\]

Key Concepts

Row-Echelon Form

In matrix algebra, determining the row-echelon form of a matrix is a foundational concept. A matrix is said to be in row-echelon form when it fulfills specific characteristics:

• All nonzero rows appear above any rows that consist entirely of zeroes.
• The leading entry, which is the first nonzero number in a row, is positioned to the right of any leading entries in the rows above.
• All entries in any column below a leading entry must be zero.

For the given matrix \( \begin{bmatrix} 1 & 0 & -3 \ 0 & 1 & 5 \end{bmatrix} \), we can see that:

• Both rows have nonzero leading entries, and there are no rows of zeroes.
• The leading entry in the first row is 1, and in the second row, it is again 1 but positioned further to the right.
• There are no entries below the leading 1s in the columns.

Therefore, this matrix satisfies all conditions and is, indeed, in row-echelon form.

Reduced Row-Echelon Form

The reduced row-echelon form heightens the conditions for a matrix beyond what is needed for row-echelon form:

• It meets all the requirements for row-echelon form.
• The leading entry in each row is exactly 1.
• Each leading 1 is the only nonzero entry in its column.

Now, examining our matrix, \( \begin{bmatrix} 1 & 0 & -3 \ 0 & 1 & 5 \end{bmatrix} \):

• The leading entries in both rows are 1, which aligns with the condition of reduced row-echelon form.
• For each leading 1, the corresponding column contains only 0s apart from the leading 1 itself.

Hence, the matrix is not only in row-echelon form but also fulfills the criteria for reduced row-echelon form. This elevated form simplifies linear algebra operations, as it often directly reveals solutions in the context of solving systems of equations.

System of Equations

An augmented matrix is often used to represent a system of equations. Each row in the matrix corresponds to an equation, and each column corresponds to the coefficients of the variables, with the last column representing the constants of the equations. To transform the given matrix into a system of equations, we look at each row as follows:

• The first row \( [1, 0, -3] \) translates to the equation: \( 1x + 0y = -3 \), simplifying to \( x = -3 \).
• The second row \( [0, 1, 5] \) translates to the equation: \( 0x + 1y = 5 \), simplifying to \( y = 5 \).

Therefore, the system of equations represented by the matrix is:\[\begin{cases} x = -3 \ y = 5 \end{cases} \]This approach allows us to easily interpret the matrix as a set of linear equations, providing a clear path to solving them for variable values.