Problem 13 - Differential Equations and Linear Algebra

Question

Express the matrix \(A\) as a product of elementary matrices. $$A=\left[\begin{array}{lll}1 & 2 & 3 \\\0 & 8 & 0 \\\3 & 4 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix A can be expressed as a product of elementary matrices: \[A = E_6E_5E_4E_3E_2E_1 = \left[\begin{array}{lll}1 & 0 & -3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \left[\begin{array}{lll}1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \frac{-1}{4} \end{array}\right] \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \frac{1}{8} & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ -3 & 0 & 1 \end{array}\right]\]

Step 1: Write down the given matrix A

First, let's write down the given matrix \(A\): \[A= \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 8 & 0 \\ 3 & 4 & 5\end{array}\right]\]

Step 2: Bring the matrix A to the identity matrix using row operations

A series of row operations on the matrix A will be used to convert A into the identity matrix: 1. Add -3 times the first row (R1) to the third row (R3): \(R_3 = R_3 - 3R_1\) \[A_1 = \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 8 & 0 \\ 0 & -2 & -4\end{array}\right]\] 2. Multiply the second row (R2) by 1/8: \(R_2= \frac{1}{8}R_2\) \[A_2 = \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 &0 \\ 0 & -2 & -4\end{array}\right]\] 3. Add 2 times the second row (R2) to the third row (R3): \(R_3 = R_3 + 2R_2\) \[A_3 = \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 &0 \\ 0 & 0 & -4\end{array}\right]\] 4. Multiply the third row (R3) by -1/4: \(R_3 = \frac{-1} {4}R_3\) \[A_4 = \left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\] 5. Add -2 times the second row (R2) to the first row (R1): \(R_1 = R_1 - 2R_2\) \[A_5 = \left[\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\] 6. Add -3 times the third row (R3) to the first row (R1): \(R_1 = R_1 - 3R_3\) \[A_6 = I = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0& 1\end{array}\right]\]

Step 3: Apply row operations to the identity matrix

Now, we will perform the same row operations to the identity matrix \(I\): 1. Define the elementary matrices corresponding to the operations: \(E_1 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\-3 & 0 & 1\end{array}\right]\), \(E_2 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & \frac{1}{8} & 0 \\0 & 0 & 1\end{array}\right]\), \(E_3 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 2 & 1\end{array}\right]\), \(E_4 = \left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\0 & 0 & \frac{-1}{4}\end{array}\right]\), \(E_5 = \left[\begin{array}{lll}1 & -2 & 0 \\ 0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\), and \(E_6 = \left[\begin{array}{lll}1 & 0 & -3 \\ 0 & 1 & 0 \\0 & 0 & 1\end{array}\right]\). 2. Compute the product of these elementary matrices: \[A=(E_6E_5E_4E_3E_2E_1)I\] Thus, matrix A can be expressed as a product of elementary matrices: \[A = E_6E_5E_4E_3E_2E_1\]

Key Concepts

Matrix Decomposition

Matrix decomposition is a powerful tool in linear algebra that helps in understanding complex matrices by breaking them down into products of simpler matrices. One common type of matrix decomposition is expressing a matrix as a product of elementary matrices.

• Elementary Matrices: Elementary matrices are derived from the identity matrix by performing a single row operation. Row operations include row swapping, scaling, and adding multiples of one row to another. These operations are reversible, making elementary matrices easy to manipulate.
• Benefits of Matrix Decomposition: Decomposing a matrix provides great insight into its structure, making operations like matrix inversion and solving linear equations more efficient. Decomposition also simplifies matrix computations by dividing complex tasks into simpler problems.

In our exercise, the decomposition of matrix \(A\) as a product of elementary matrices simplifies the transformation of \(A\) into the identity matrix \(I\) through a sequence of row operations. By identifying and applying the relevant elementary matrices, we achieve a straightforward matrix manipulation process.

Row Operations

Row operations are fundamental tools in matrix manipulation, allowing the transformation of a matrix into more useful forms, such as the identity matrix. They are also key elements in matrix decomposition.Three main types of row operations exist:

• Row Swapping: This involves exchanging two rows of the matrix. It helps in repositioning a matrix element to a desired location, aiding in matrix simplification.
• Row Scaling: Here, a row is multiplied by a non-zero scalar. It is used to normalize a row or simplify its elements.
• Row Addition (Replacement): This operation adds or subtracts a multiple of one row from another. It is crucial for eliminating elements and creating zeros in strategic positions in a matrix.

In the given exercise, row operations were applied systematically to transform matrix \(A\) to the identity matrix \(I\) by eliminating non-diagonal elements step by step, showcasing their ability to simplify and solve matrix-related problems.

Identity Matrix

The identity matrix is a special matrix that acts like the number 1 in matrix multiplication. It's crucial for understanding elementary matrices and operations like matrix decomposition.

• Definition: An identity matrix, denoted by \(I\), is a square matrix with ones on the diagonal from the top left to the bottom right and zeros elsewhere.
• Properties: The identity matrix maintains the original matrix when multiplied by it. This property is vital for confirming matrix transformations and ensuring calculations are consistent.

In the context of decomposing a matrix into elementary matrices, the identity matrix serves as the target transformation. Starting from matrix \(A\), the goal is to apply row operations until \(A\) is reshaped into \(I\), confirming the sequence of elementary operations was performed accurately. Understanding the identity matrix is key to grasping matrix operations, as it often serves as the goal post in matrix manipulation tasks.