Question

Find the rank of each of the following matrices. $$ \left(\begin{array}{rrrrr} 2 & 2 & 8 & 6 & 2 \\ -1 & -2 & -1 & 0 & -1 \\ 4 & 6 & 13 & 9 & 4 \\ -4 & -8 & -16 & -12 & -4 \end{array}\right) $$

Short answer

The rank of the matrix is 3.

Step-by-step solution

- Identify the matrix

Let's denote the given matrix as matrix A. The matrix A is:

- Write down matrix A

\[A = \left( \begin{array}{rrrrr}2 & 2 & 8 & 6 & 2 \-1 & -2 & -1 & 0 & -1 \4 & 6 & 13 & 9 & 4 \-4 & -8 & -16 & -12 & -4\end{array} \right)\]

- Perform row operations to transform matrix A into row-echelon form

First, add the second row to the first row: \[R1 = R1 + R2\]The matrix becomes: \[\left( \begin{array}{rrrrr}1 & 0 & 1 & 2 & 1 \-1 & -2 & -1 & 0 & -1 \4 & 6 & 13 & 9 & 4 \-4 & -8 & -16 & -12 & -4\end{array} \right)\]

- Simplify the matrix further using row operations

Add 4 times the first row to the fourth row: \[R4 = R4 + 4R1\]The matrix becomes: \[\left( \begin{array}{rrrrr}1 & 0 & 1 & 2 & 1 \-1 & -2 & -1 & 0 & -1 \4 & 6 & 13 & 9 & 4 \0 & -8 & -12 & -4 & 0\end{array} \right)\]

- Use more row operations to get more zeros

Add the second row to the third row and then to the last row: \[R3 = R3 + R2\]\[R4 = R4 + R2\]This simplifies the matrix to: \[\left( \begin{array}{rrrrr}1 & 0 & 1 & 2 & 1 \-1 & -2 & -1 & 0 & -1 \3 & 4 & 12 & 9 & 3 \1 & -10 & -13 & 0 & -1\end{array} \right)\]

- Get the matrix in row-echelon form

To attain row-echelon form, we need the leading non-zero entry in each row to be 1 and for each leading 1 to have 0 in every row below it: \[\left( \begin{array}{rrrrr}1 & 0 & 1 & 2 & 1 \0 & -2 & -2 & 2 & 0 \0 & 4 & 11 & 0 & 0 \0 & 0 & 0 & 1 & 1\end{array} \right)\]

- Determine the rank

The rank of a matrix is equal to the number of non-zero rows. Here, there are 3 non-zero rows in the row-echelon form of the matrix.

Key concepts

Row-Echelon Form

To find the rank of a matrix, we often transform it into row-echelon form. This form requires that each row begins with a leading 1, and every row below a leading 1 consists of zeros in the respective column. Converting a matrix to row-echelon form simplifies the process of determining its rank, which corresponds to the number of non-zero rows. Row-echelon form is essential for solving systems of linear equations and understanding the row space of a matrix. Always remember, achieving row-echelon form involves using row operations such as row swaps, scalar multiplication, and adding multiples of rows to other rows.

Linear Algebra

Linear algebra is a fundamental area of mathematics that deals with vectors, vector spaces, and linear transformations. Matrices, as arrays of numbers, are pivotal in this field. They represent linear transformations and can be manipulated using various algebraic operations. Understanding matrices, including concepts like rank, row-echelon form, and row operations, is crucial in linear algebra. These concepts allow us to solve systems of linear equations, study vector spaces, and perform various mathematical transformations and decompositions.

Row Operations

Row operations are key tools used to manipulate matrices. They include three primary types: swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another. These operations help us transform a matrix into simpler forms like row-echelon form or reduced row-echelon form without changing its rank. Mastering row operations is essential for tasks such as solving linear equations and determining matrix rank. Always perform row operations systematically to avoid errors and ensure the transformations are accurate.

Non-Zero Rows

Non-zero rows in a matrix are those that contain at least one non-zero element. When a matrix is in row-echelon form, non-zero rows help determine its rank. The rank is the number of non-zero rows in this form. Identifying and counting these rows give insight into the linear independence of the rows and the dimension of the row space. Non-zero rows are crucial in understanding the structure and properties of a matrix. Keep track of transformations to accurately count non-zero rows when evaluating rank.