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Chapter 7: Problem 180

Let \(\mathrm{A}\) be the matrix $$ \begin{array}{rrrrrr} \mid 0 & 1 & 3 & -2 & -1 & 2 \\ 10 & 2 & 6 & -4 & -2 & 4 \\ \mid 0 & 1 & 3 & -2 & 1 & 4 \\ \mid 0 & 2 & 6 & 1 & -1 & 0 \end{array} $$ Find the determinant rank of \(\mathrm{A}\)

Short Answer

Expert verified
The determinant rank of matrix A is \(4\).

Step by step solution

01

Create a row echelon form for matrix A

First, let's write down the given matrix A: \[ A= \begin{bmatrix} 0 & 1 & 3 & -2 & -1 & 2\\ 10 & 2 & 6 & -4 & -2 & 4\\ 0 & 1 & 3 & -2 & 1 & 4\\ 0 & 2 & 6 & 1 & -1 & 0 \end{bmatrix} \] Now, perform row operations to achieve row echelon form: 1. Swap row 1 and row 2 so that the first row has a non-zero entry. 2. Subtract 2 * row 1 from row 4: \[ A= \begin{bmatrix} 10 & 2 & 6 & -4 & -2 & 4\\ 0 & 1 & 3 & -2 & -1 & 2\\ 0 & 1 & 3 & -2 & 1 & 4\\ 0 & 0 & 0 & 9 & 5 & -8 \end{bmatrix} \] 3. Add row 2 to row 3: \[ A= \begin{bmatrix} 10 & 2 & 6 & -4 & -2 & 4\\ 0 & 1 & 3 & -2 & -1 & 2\\ 0 & 0 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 9 & 5 & -8 \end{bmatrix} \] 4. Swap row 3 and row 4: \[ A= \begin{bmatrix} 10 & 2 & 6 & -4 & -2 & 4\\ 0 & 1 & 3 & -2 & -1 & 2\\ 0 & 0 & 0 & 9 & 5 & -8\\ 0 & 0 & 0 & 0 & 0 & 2 \end{bmatrix} \] Now, we have achieved the row echelon form.
02

Determine the determinant rank

The determinant rank of a matrix is the number of non-zero rows in its row echelon form. Counting the non-zero rows in the row echelon form of matrix A, we obtain: Determinant rank of matrix A = 4. So, the determinant rank of matrix A is 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Row Echelon Form
Understanding the row echelon form of a matrix is essential for various calculations in linear algebra, including finding the determinant and rank. This form simplifies a matrix into a shape that is much easier to work with.

To convert a matrix into its row echelon form, we apply a process known as Gaussian elimination. This process involves a series of row operations, including swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another.

The rules for a matrix to be in row echelon form are as follows:
  • All nonzero rows are above any rows of all zero.
  • Each leading entry (or pivot) in a row is to the right of the leading entry in the row above it.
  • All entries in a column below a leading entry are zeros.
Once in this form, the matrix's structure makes it clear how to proceed with other operations, such as determining the rank or solving systems of linear equations.
Matrix Rank

Importance of Matrix Rank

In linear algebra, the rank of a matrix is a fundamental concept that is indicative of the matrix's properties. It tells us the maximum number of linearly independent column vectors in the matrix or equivalently, the maximum number of linearly independent row vectors.

The rank is important because it provides information about the solutions to systems of linear equations related to the matrix. A higher rank means more unique solutions, while a rank deficient matrix could have no solution or an infinite number of solutions.

Determining Rank Using Row Echelon Form

We can find the rank of a matrix by transforming it into its row echelon form. Once in this form, the rank is simply the number of non-zero rows. This method simplifies the tedious process that might involve working with full row operations and allows for an easy count of the linearly independent rows.
Row Operations
The manipulation of a matrix using row operations plays a crucial role in several areas of linear algebra and is key to simplifying matrices. There are three fundamental row operations:
  • Row Switching: Swapping two rows of a matrix. This operation does not affect the determinant of the matrix.
  • Row Multiplication: Multiplying all entries of a row by a non-zero scalar. This changes the determinant of the matrix by the scalar by which the row was multiplied.
  • Row Addition: Adding or subtracting a multiple of one row to another row. This operation helps to create zeros in a row, making it easier to form the row echelon shape.
Row operations are essential for various algorithms in linear algebra, such as finding inverses, solving systems of equations, and determining determinants and rank as seen in the exercise.

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Most popular questions from this chapter

Show that the matrix $$ \mathrm{A}=\begin{array}{cll} 0 & 1 & 2 \\ 2 & 3 & 4 \\ 14 & 7 & 10 \end{array} $$ is equivalent to \(\mathrm{D}^{33}{ }_{2}\) where \(\mathrm{D}^{\mathrm{m}, \mathrm{n}} \mathrm{r}\) denotes the canonical form under equivalence of A. D \(^{\mathrm{m}, \mathrm{n}}_{\mathrm{r}}\) is the echelon form that has one's along the diagonal and zeros elsewhere, and where all the zero rows are consigned to the depths of the matrix.

Let the homogeneous linear system \(\mathrm{AX}=\mathrm{B}\) be given by \(\begin{array}{llll}1 & 2 & 0 \mid & \left|x_{1}\right| & |0|\end{array}\) \(|0 \quad 1 \quad 3| \quad\left|\mathrm{x}_{2}\right|=|0|\) \(\mid \begin{array}{llll}2 & 1 & 3 \mid & \left|\mathrm{x}_{3}\right| & |0| \text { . }\end{array}\) Show that \(\mathrm{A}\) has only the trivial solution, \(\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=(0,0,0)\)

Find the rank of matrix A where $$ \mathrm{A}=\begin{array}{cccc} 1 & 0 & 2 & 3 \mid \\ 0 & 0 & 5 & 1 \\ 0 & 0 & 0 & 0 \end{array} $$

Give an example to illustrate the following theorem: The system of n homogeneous linear equations in \(\mathrm{n}\) unknowns, \(\mathrm{AX}=0\), has a nontrivial solution if and only if rank \(\mathrm{A}<\mathrm{n}\)

Let \(\mathrm{P}_{4}\), denote the vector space of all polynomials of degree at most equal to four. Let \(\mathrm{V}\) be the subspace of \(\mathrm{P}_{4}\). spanned by \(\mathrm{S}=\left\\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right\\}\) where \(\alpha_{1}=\mathrm{t}^{4}+\mathrm{t}^{2}+2 \mathrm{t}+1\) \(\alpha_{2}=t^{4}+t^{2}+2 t+2, \alpha_{3}=2 t^{4}+t^{3}+t+2\) and \(\alpha_{4}=t^{4}+t^{3}-t^{2}\) \(-\mathrm{t}\). Find a basis for \(\mathrm{V}\).
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