Question

In each case find an elementary matrix \(E\) such that \(B=E A\). a. \(A=\left[\begin{array}{rr}2 & 1 \\ 3 & -1\end{array}\right], B=\left[\begin{array}{rr}2 & 1 \\ 1 & -2\end{array}\right]\) b. \(A=\left[\begin{array}{rr}-1 & 2 \\ 0 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right]\) c. \(A=\left[\begin{array}{rr}1 & 1 \\ -1 & 2\end{array}\right], B=\left[\begin{array}{rr}-1 & 2 \\ 1 & 1\end{array}\right]\) d. \(A=\left[\begin{array}{ll}4 & 1 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rr}1 & -1 \\ 3 & 2\end{array}\right]\) e. \(A=\left[\begin{array}{rr}-1 & 1 \\ 1 & -1\end{array}\right], B=\left[\begin{array}{ll}-1 & 1 \\ -1 & 1\end{array}\right]\) f. \(A=\left[\begin{array}{rr}2 & 1 \\ -1 & 3\end{array}\right], B=\left[\begin{array}{rr}-1 & 3 \\ 2 & 1\end{array}\right]\)

Short answer

For each exercise, find the elementary matrix by determining and applying the appropriate row operation.

Step-by-step solution

Understand the concept of elementary matrices

Elementary matrices are derived from the identity matrix by performing a single elementary row operation. These operations include swapping two rows, multiplying a row by a non-zero constant, or adding a multiple of one row to another row.

Exercise a: Identify row operation

Comparing the matrices, we see the second row of matrix B, \([1, -2]\), is obtained by replacing the second row of matrix A with itself minus two times the first row. So, the operation needed is: \(R_2 - 2R_1 \rightarrow R_2\).

Exercise a: Construct elementary matrix

Start with the identity matrix \(I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). To perform the operation \(R_2 - 2R_1 \rightarrow R_2\), we modify the identity matrix to be \(E = \begin{bmatrix} 1 & 0 \ -2 & 1 \end{bmatrix}\). Multiply this by A to get B.

Exercise b: Identify row operation

Looking at matrices A and B, the first row of B is the opposite of the first row of A. Therefore, the operation is multiplying the first row by -1: \(-R_1 \rightarrow R_1\).

Exercise b: Construct elementary matrix

To reflect \(-R_1 \rightarrow R_1\), modify the identity matrix to \(E = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}\). Multiply E by A to obtain B.

Exercise c: Identify row operations

By comparing the matrices, it is evident that both rows have swapped places: \(R_1 \leftrightarrow R_2\).

Exercise c: Construct elementary matrix

To achieve the row swap, change the identity matrix to \(E = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\). Multiplying this with A gives B.

Exercise d: Identify row operations

The first row is modified into 1/4th of its original: \((1/4)R_1 \rightarrow R_1\).

Exercise d: Construct elementary matrix

Start with identity and adjust to reflect the operation: \(E = \begin{bmatrix} 1/4 & 0 \ 0 & 1 \end{bmatrix}\). Multiply E by A to achieve B.

Exercise e: Identify row operation

The second row of B is the negative of the second row of A: \(-R_2 \rightarrow R_2\).

Exercise e: Construct elementary matrix

Make the modification in \(I_2\): \(E = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}\). Use this to transform A into B.

Exercise f: Identify row operations

In B, rows of A are swapped: \(R_1 \leftrightarrow R_2\). This represents an elementary transformation on the identity matrix.

Exercise f: Construct elementary matrix

To swap the rows, the elementary matrix is \(E = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix}\). Applying this to A gives B.

Key concepts

Row Operations

Row operations are essential transformations that you can apply to matrices in linear algebra. They help in solving systems of linear equations and facilitate matrix manipulations. There are three types of elementary row operations: switching two rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another row. Each of these operations can be represented as an elementary matrix when applied to an identity matrix. This makes it easy to apply the same operation to any other matrix by multiplication.
To understand how these operations work with the exercise provided, consider that for each matrix transformation from matrix A to matrix B, a specific row operation was applied, creating an elementary matrix E. These operations might involve swapping rows, which changes the position of each row, scaling a row, which changes the values of all elements within that row by a single factor, or row replacement/addition operation, which involves replacing a row by the sum of itself and a multiple of another row.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce a third matrix. The number of columns in the first matrix must equal the number of rows in the second matrix to carry out the multiplication. Each element of the resulting matrix is calculated by taking the dot product of the corresponding row of the first matrix and the column of the second.
When you multiply an elementary matrix with another matrix, the result will perform the row operation on the matrix. In the exercises, each elementary matrix E represents a particular row operation. When multiplying E by matrix A, you derive matrix B. This demonstrates how matrix multiplication with elementary matrices can transform one matrix into another by applying systematic row operations.

Identity Matrix

The identity matrix is a special type of matrix in linear algebra. It is square-shaped, meaning it has the same number of rows and columns, and it contains ones on its main diagonal (from top-left to bottom-right) and zeros elsewhere. For example, a 2x2 identity matrix is: \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
In the context of elementary matrices, the identity matrix acts as a neutral or identity element in matrix multiplication. This means that when any matrix, A, is multiplied by the identity matrix, it remains unchanged. However, when you alter the identity matrix by performing a row operation, you create an elementary matrix. These elementary matrices used above were derived directly from an identity matrix, indicating the specific row operations necessary to transform matrix A into matrix B through multiplication (e.g., row swaps, row scaling, or adding a row to another row).

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations between these structures. It is widely used not just in mathematics, but also in physics, engineering, computer science, and economics. The study of linear transformations involves understanding matrices, which provide a powerful and efficient tool to represent and manipulate linear transformations.
Within linear algebra, concepts such as elementary matrices, matrix multiplication, and identity matrices are fundamental in simplifying systems of equations, solving them effectively, and understanding vector space theory. The process of using elementary matrices to perform specific row operations gives insight into solving systems of linear equations and understanding better numerical methods and algorithms in computational mathematics.