Question

For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse. a. \(E=\left[\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\) b. \(E=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]\) c. \(E=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 1\end{array}\right]\) d. \(E=\left[\begin{array}{rrr}1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\) e. \(E=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]\) f. \(E=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5\end{array}\right]\)

Short answer

a. Add 3 times R3 to R1, inverse: subtract; b. Swap R1 and R3, same inverse; c. Multiply R2 by 0.5, inverse: by 2; d. Add 2 times R1 to R2, inverse: subtract; e. Swap R1 and R2, same inverse; f. Multiply R3 by 5, inverse: by 0.2.

Step-by-step solution

Analyze Matrix a

Matrix a is \( E = \begin{bmatrix} 1 & 0 & 3 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). This matrix adds 3 times the third row to the first row. The inverse operation is to subtract 3 times the third row from the first row. Thus, the inverse matrix is \( E^{-1} = \begin{bmatrix} 1 & 0 & -3 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Analyze Matrix b

Matrix b is \( E = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} \). This matrix swaps the first and third rows. Swapping the same rows again will return to the original matrix, so the inverse is identical to the matrix itself: \( E^{-1} = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} \).

Analyze Matrix c

Matrix c is \( E = \begin{bmatrix} 1 & 0 & 0 \ 0 & \frac{1}{2} & 0 \ 0 & 0 & 1 \end{bmatrix} \). This operation multiplies the second row by \( \frac{1}{2} \). The inverse operation is to multiply the second row of the inverse matrix by 2, so the inverse matrix is \( E^{-1} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Analyze Matrix d

Matrix d is \( E = \begin{bmatrix} 1 & 0 & 0 \ -2 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). This matrix adds 2 times the first row to the second row. The inverse operation subtracts 2 times the first row from the second row. Therefore, the inverse matrix is \( E^{-1} = \begin{bmatrix} 1 & 0 & 0 \ 2 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Analyze Matrix e

Matrix e is \( E = \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} \). This matrix swaps the first and second rows. Similar to matrix b, swapping the same rows again returns the original arrangement, so the inverse is the same as the matrix: \( E^{-1} = \begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Analyze Matrix f

Matrix f is \( E = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 5 \end{bmatrix} \). This matrix multiplies the third row by 5. The inverse operation divides the third row by 5. Thus, the inverse matrix is \( E^{-1} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & \frac{1}{5} \end{bmatrix} \).

Key concepts

Row Operations

Elementary matrices are essential tools in performing row operations on matrices. These operations include row swapping, row multiplication, and row addition. Each operation can be represented by an elementary matrix. When you multiply a matrix by an elementary matrix, you're essentially applying that row operation to it.

For instance, the matrix \( E = \begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix} \) represents swapping the first and third rows of another matrix. Such straightforward manipulations help us simplify matrices and solve systems of equations efficiently. Understanding these operations is key to mastering linear algebra concepts and working with more complex matrix equations.

Matrix Inversion

The process of finding the inverse of a matrix involves determining another matrix which, when multiplied by the original, yields the identity matrix. Not all matrices have inverses, but elementary matrices do, thanks to their simple nature.

For elementary matrices, the inverse is found by reversing the row operation they represent. For example, if an elementary matrix represents an operation of multiplying a row by a scalar, its inverse would involve multiplying that row by the reciprocal of the scalar. This isn't just a neat mathematical curiosity; knowing inverses helps when solving linear systems. If a system can be expressed as \( AX = B \), you can find \( X \) using the inverse of \( A \) if it exists, through \( X = A^{-1}B \). This process highlights the value of simplicity that elementary matrices provide in finding inverses.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. Its core involves solving linear equations and understanding vector transformations.

Elementary matrices and row operations are powerful techniques within linear algebra. They are used to alter a matrix to its row echelon or reduced row echelon form, which is crucial in solving linear systems. By using these techniques, complex linear equations become more straightforward to resolve.

This field extends beyond just equation solving. Its concepts underpin vast applications in computer science, physics, and engineering. Mastery of linear algebra comes with understanding how matrices and systems interact, made simpler through elementary matrices and their operations.

Matrix Transformations

Matrix transformations convert vectors in one space to vectors in another. This is a primary application of matrix multiplication. Essentially, matrices can define transformations like rotations, scaling, and translations in vector spaces.

Elementary matrices are foundational in thinking about these transformations. Each elementary transformation corresponds to a basic geometric change.

• Row swaps in a matrix can be associated with reflecting vector positions.
• Row scaling relates to resizing or scaling transformations.
• Addition or subtraction operations transform shapes by shifting components accordingly.

Understanding matrix transformations involves seeing how these actions affect vector positions and orientations, a vital skill in graphics and data analysis. Elementary matrices help simplify and illustrate more complex transformations by breaking them down into manageable operations.