Problem 24 - Precalculus

Question

Indicate whether each matrix is in row-echelon form. If it is, determine whether it is in reduced row-echelon form. $$\left[\begin{array}{lll|l} 1 & 0 & 0 & -3 \\ 0 & 1 & 3 & 14 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix is in reduced row-echelon form.

Step 1: Understand Row-Echelon Form

In row-echelon form, each non-zero row begins with a 1 (called a leading 1). Any rows consisting entirely of zeros are at the bottom. The leading 1 in each row is to the right of the leading 1 in the row above. Each leading 1 is the only non-zero entry in its column from the row it's in and above.

Step 2: Check Given Matrix for Row-Echelon Form

Examine the matrix:\[\begin{array}{ccc|c} 1 & 0 & 0 & -3 \ 0 & 1 & 3 & 14 \end{array}\]Assess the matrix against the properties of row-echelon form. Here, the first row's leading entry is 1, and the leading entry of the second row is also 1, which is to the right of the leading entry in the first row. Hence, the matrix is in row-echelon form.

Step 3: Understand Reduced Row-Echelon Form

In reduced row-echelon form, the matrix must firstly satisfy all the conditions for row-echelon form. Additionally, every leading 1 is the only non-zero entry in its column. Each column that contains a leading 1 must have all other entries equal to 0.

Step 4: Check Given Matrix for Reduced Row-Echelon Form

Check if each leading 1 is the only non-zero entry in its column: - In the first column, the leading 1 at position (1,1) is the only non-zero entry. - In the second column, the leading 1 at position (2,2) is only for the second row and the first row has zero in this position, so it is the only non-zero entry. All these conditions are satisfied, so the matrix is in reduced row-echelon form.

Key Concepts

Row-Echelon Form

When working with matrices in linear algebra, understanding different forms such as the row-echelon form is essential. Matrices in row-echelon form have distinct characteristics that make them easier to handle, especially in solving systems of linear equations. Here are the key properties:

• Any row consisting entirely of zeros is at the bottom of the matrix.
• The first non-zero number from the left in a non-zero row is a 1, known as a leading 1.
• The leading 1 in any row is positioned to the right of the leading 1 in the row directly above it.

Checking a matrix for row-echelon form involves confirming these conditions. In the matrix provided in the exercise, each row has a leading 1 that follows these rules, making the matrix in row-echelon form.

Matrix Algebra

Matrix algebra is an extension of elementary algebra, primarily designed to manipulate matrices. Matrices are grid-like structures filled with numerical values arranged in rows and columns. Understanding operations such as addition, multiplication, and scalar multiplication are crucial in matrix algebra.
In analyzing matrices, the concept of row-echelon form comes into play to simplify matrices for easier manipulation. Matrix algebra uses row operations, which are:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting the multiple of one row from another.

These operations make it possible to transform any matrix into a row-echelon form, facilitating simpler matrix breakup for problem-solving. This process is vital in dealing with complex computations in linear algebra.

Linear Algebra

Linear algebra is a fundamental area of mathematics with applications in various domains such as physics, computer science, and engineering. It deals with vectors, spaces, and linear transformations. One of the primary objects of study in linear algebra is matrices, which represent linear transformations in space.
In linear algebra, systems of linear equations can be represented efficiently using matrices. Putting matrices into forms like row-echelon and reduced row-echelon makes it easier to analyze and solve these systems.
Additionally, linear algebra is pivotal in determining solutions to linear systems through concepts like matrix decompositions, vector spaces, and eigenvalues. A proper understanding of matrices and how to manipulate them using various forms and operations is key to excelling in linear algebra and applying it to solve real-world problems.