Question

Bestimmen Sie den Rang \(r\) der folgenden Matrizen: $$ \mathrm{A}=\left(\begin{array}{rrrr} -1 & 4 & 1 & 3 \\ 2 & -2 & -2 & 0 \\ 0 & 2 & 0 & 2 \end{array}\right) \quad \mathrm{B}=\left(\begin{array}{llll} 3 & 2 & 2 & 2 \\ 4 & 2 & 4 & 2 \\ 3 & 1 & 3 & 1 \\ 2 & 1 & 2 & 1 \end{array}\right) $$

Short answer

The rank of Matrix A is 3 and the rank of Matrix B is 1.

Step-by-step solution

Gaussian Elimination for Matrix A

Start by writing the given Matrix A and using Gaussian elimination to bring it to row-echelon form. That can be done by swapping, scaling, or adding rows, until all zeros are below each leading one (the first non-zero number in each row, moving from top to bottom). Here we want the top-left corner to be a 1 for simplicity, so we can swap row 1 with row 2. \[A=\left(\begin{array}{rrrr} 2 & -2 & -2 & 0 \ -1 & 4 & 1 & 3 \ 0 & 2 & 0 & 2 \end{array}\right)\] Then, you can scale row 1 by 1/2 and add row 1 to row 2 to get a 0 in the second position of the second row: \[A=\left(\begin{array}{rrrr} 1 & -1 & -1 & 0 \ 0 & 3 & 0 & 3 \ 0 & 2 & 0 & 2 \end{array}\right)\] This is now in row-echelon form.

Finding Rank of Matrix A

Now, look at the row-echelon form of Matrix A: \[A=\left(\begin{array}{rrrr} 1 & -1 & -1 & 0 \ 0 & 3 & 0 & 3 \ 0 & 2 & 0 & 2 \end{array}\right)\] Any row of zeros would be disregarded when counting the rank. Here, no rows are all zeroes, so the rank of this matrix is 3, the number of rows.

Gaussian Elimination for Matrix B

Similarly, for Matrix B, we carry out the same process of Gaussian elimination to get it in its row-echelon form. \[B=\left(\begin{array}{llll} 3 & 2 & 2 & 2 \ 4 & 2 & 4 & 2 \ 3 & 1 & 3 & 1 \ 2 & 1 & 2 & 1 \end{array}\right)\] The row-echelon form of Matrix B is: \[B=\left(\begin{array}{llll} 1 & 2/3 & 1 & 2/3 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)\] after swapping rows and using the first row to eliminate the lower ones.

Finding Rank of Matrix B

Looking at the row-echelon form of Matrix B we get the following: \[B=\left(\begin{array}{llll} 1 & 2/3 & 1 & 2/3 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{array}\right)\] Here, only the first row isn't all zeroes, so the rank of this matrix is 1.

Key concepts

Gaussian Elimination

Gaussian Elimination is a method used in Linear Algebra to solve systems of linear equations and find the rank of a matrix. It involves performing operations on the rows of an augmented matrix to bring it into a simpler form called row-echelon form. In Gaussian Elimination:

• We can swap two rows to reorder them for easier simplification.
• We are allowed to multiply all elements of a row by a non-zero constant.
• We can add or subtract the multiples of one row from another row to eliminate variables.

These operations aim to create leading 1s (the first non-zero number in each row) and to make zeros below these leading 1s. By simplifying the system of equations, Gaussian Elimination helps us find solutions more easily and also enables finding the matrix rank, as it brings about the fundamental rows that define the solution space.

Row Echelon Form

Row Echelon Form (REF) is an important concept in Linear Algebra and serves as an intermediate step in solving matrix problems such as determining rank. A matrix is in row-echelon form when:

• Every non-zero row has a leading 1 position further to the right with each subsequent row.
• All the elements below a leading 1 are zeros.
• Any all-zero rows are at the bottom of the matrix if they exist.

Achieving row-echelon form involves making strategic row operations like scaling, row swapping, and row additions. For example, in the process of determining the rank of Matrix A, moving row positions and scaling were key steps to make the leading numbers clear. Row Echelon Form effectively categorizes the significant rows of a matrix, simplifying finding properties like rank directly.

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces, systems of linear equations, and matrices. It forms the foundational language and framework for a wide variety of complex systems and problem-solving in fields such as physics, engineering, computer science, and economics.

• Matrices: Rectangular arrays of numbers that can represent coefficients in systems of linear equations.
• Vectors: Objects representing a direction and magnitude in space, often used in Linear Algebra to solve systems and transformations.
• Rank: The dimension of the vector space spanned by the rows (or columns) of a matrix, representing the number of independent rows or columns.

By understanding Linear Algebra, one can grasp how toolsets like Gaussian Elimination and transformations into Row Echelon Form work in practice. It's all about solving systems to find unknowns and understanding underlying structures in mathematical data.