Problem 46 - Differential Equations and Linear Algebra

Question

Reduce the matrix \(\left[A \quad I_{n}\right]\) to reduced row-echelon form and thereby determine, if possible, the inverse of \(A\). \(A\) is a randomly generated \(4 \times 4\) matrix.

Step-by-Step Solution

Answer

In conclusion, after forming the augmented matrix \([A \ | \ I_{4}]\), we perform row operations to reduce it to a reduced row-echelon form. If the left part (originally matrix A) is transformed into an identity matrix, then matrix A is invertible, and its inverse can be read off from the right part of the reduced matrix: \[A^{-1} = \begin{pmatrix} a' & b' & c' & d' \\ e' & f' & g' & h' \\ i' & j' & k' & l' \\ m' & n' & o' & p'\end{pmatrix}\] Otherwise, matrix A is not invertible.

Step 1: Form the augmented matrix

Concatenate the given matrix A with a 4x4 identity matrix: \[[ A \ | \ I_{4}] = \left[\begin{array}{cccc|cccc} a & b & c & d & 1 & 0 & 0 & 0 \\ e & f & g & h & 0 & 1 & 0 & 0 \\ i & j & k & l & 0 & 0 & 1 & 0 \\ m & n & o & p & 0 & 0 & 0 & 1 \end{array}\right]\] Next, perform Gaussian elimination and row operations to reduce this augmented matrix into a reduced row-echelon form.

Step 2: Perform row operations

Apply row operations, including row interchange, row scaling, and row addition to reduce the augmented matrix to its row-echelon form: \[[A \ | \ I_{4}] \xrightarrow[]{rref} \left[\begin{array}{cccc|cccc} 1 & 0 & 0 & 0 & a' & b' & c' & d' \\ 0 & 1 & 0 & 0 & e' & f' & g' & h' \\ 0 & 0 & 1 & 0 & i' & j' & k' & l' \\ 0 & 0 & 0 & 1 & m' & n' & o' & p' \\ \end{array}\right]\] where rref stands for Reduced Row-Echelon Form and a', b', c', d', e', f', g', h', i', j', k', l', m', n', o', and p' are new numbers after performing row operations. Now, observe the transformed form of the augmented matrix.

Step 3: Check for invertibility

If the left part of the reduced matrix (originally matrix A) has been transformed into a 4x4 identity matrix, then matrix A is invertible, and the right side of the reduced matrix will be the inverse of matrix A: \[[A^{-1}] = \begin{pmatrix} a' & b' & c' & d' \\ e' & f' & g' & h' \\ i' & j' & k' & l' \\ m' & n' & o' & p'\end{pmatrix}\] However, if the left part of the reduced matrix is not an identity matrix, then matrix A is not invertible. In conclusion, we can determine the invertibility of a given matrix A and find its inverse (if it exists) by creating an augmented matrix with an identity matrix, reducing it to the reduced row-echelon form, and then checking whether the left side of the matrix is transformed into an identity matrix. If it is, then A is invertible, and the inverse can be read off from the right part of the reduced matrix.

Key Concepts

Reduced Row-Echelon Form

When dealing with matrices, the reduced row-echelon form (RREF) is a special type of matrix form that simplifies solving linear systems. In RREF:

• Each leading entry in a row is 1, known as a leading 1.
• Leading 1s are the only non-zero entries in their column.
• Leading 1s in lower rows are to the right of those in the rows above.
• Rows with all zeros are at the bottom of the matrix.

Row operations such as row swapping, row multiplication, and row addition are used to convert a matrix to its RREF. This form helps easily identify the rank of a matrix, solutions of linear equations, and, in our context, check for invertibility.

Gaussian Elimination

Gaussian elimination is a method to solve systems of linear equations and to simplify matrices. It's named after Carl Friedrich Gauss, a renowned mathematician. This method proceeds through a series of operations:

• Use row operations to transform the matrix into upper triangular form (also known as row-echelon form).
• This means all entries below the leading coefficient (the largest absolute value in each column) are zeros.

Following these steps prepares the matrix for back substitution, where each variable is solved in reverse order. This method also sets the stage for transitioning a matrix into reduced row-echelon form, helping determine solutions or invertibility.

Augmented Matrix

An augmented matrix combines the coefficients of a system of linear equations with their corresponding constants into a single matrix. In matrix inversion:

• We take the matrix in question, say matrix A, and append an identity matrix of the same size.
• The process involves row operations that lead the original matrix part to become an identity matrix.
• If successful, the altered identity section provides the inverse of the original matrix.

Constructing an augmented matrix is a visual and structural way to handle systems of equations and explore the invertibility of matrices, allowing for direct visualization of the transformations applied.

Invertibility of a Matrix

The invertibility of a matrix refers to whether a matrix has an inverse. For a square matrix A, if there exists another matrix, A^-1, such that:

• \( A \cdot A^{-1} = I \)
• \( A^{-1} \cdot A = I \)

Where \( I \) is the identity matrix. The matrix A is then said to be invertible or non-singular.An important aspect of determining invertibility is ensuring that no zero-rows appear in what should be the identity half of the augmented matrix.If after applying row operations, the left half of the augmented matrix becomes an identity matrix, the matrix is confirmed as invertible, and its inverse is revealed on the right half.