Problem 36 - Differential Equations and Linear Algebra

Question

(a) find a row-echelon form of the given matrix \(A,\) (b) determine rank \((A),\) and (c) use the GaussJordan Technique to determine the inverse of \(A,\) if it exists. $$A=\left[\begin{array}{rrr}3 & 0 & 0 \\ 0 & 2 & -1 \\ 1 & -1 & 2\end{array}\right].$$

Step-by-Step Solution

Answer

The row-echelon form of matrix \(A\) is: \[ \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 0 \end{array}\right]. \] The rank of matrix \(A\) is 2, and the inverse of matrix \(A\) does not exist because it is not a full rank matrix.

Step 1: Rewrite the given matrix

First, rewrite the given matrix \(A\), so we can perform subsequent steps on it: \[A = \left[\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 2 & -1 \\ 1 & -1 & 2 \end{array}\right]. \]

Step 2: Find the row-echelon form

To find the row-echelon form of matrix \(A\), we perform Gaussian elimination. Here's the step-by-step process: 1. Swap rows 1 and 3 since the first element of the first row should be a non-zero entry if possible. 2. Divide row 1 by 3 to have a leading 1 in the first row. This gives us the row-echelon form of \(A\): \[A = \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 0 \end{array}\right]. \]

Step 3: Determine the rank of \(A\)

To find the rank of matrix \(A\), count the number of non-zero rows in its row-echelon form. There are 2 non-zero rows, so the rank of matrix \(A\) is 2: \[rank(A) = 2.\]

Step 4: Determine if the inverse of \(A\) exists

The inverse of a square matrix exists only if the matrix is full rank. Matrix \(A\) has a rank of 2, but it is a \(3\times3\) matrix, which means it is not full rank. Therefore, the inverse of \(A\) does not exist.

Step 5: Conclusion

Now, we can provide the final results: (a) The row-echelon form of matrix \(A\) is: \[ \left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 0 \end{array}\right]. \] (b) The rank of matrix \(A\) is 2. (c) The inverse of matrix \(A\) does not exist because it is not a full rank matrix.

Key Concepts

Row-echelon form

Matrices can look complex, but transforming them into a row-echelon form simplifies things a lot. Imagine reorganizing a room, so everything is in neat rows. That's what we do here with matrices. In row-echelon form, matrices have leading 1s, essentially the first non-zero number in a row, and zeros beneath them. This structure makes solving systems of linear equations both orderly and efficient.

For instance, converting a matrix often involves swapping rows or dividing a row by a number to get that crucial leading 1. Consider our matrix example where we swapped and divided rows to get:

• First row: leading 1 appears
• Below this row, leading numbers turn to zeros

These rules lead matrices into a tidy order, one that tells us much about their solutions!

Gaussian elimination

Gaussian elimination is like having a toolbox with Swiss Army skills! Used to simplify matrices, it involves two main tasks: turning a matrix into a row-echelon form and identifying solutions of systems of equations. Let's break it into digestible steps.

Primarily, Gaussian elimination consists of transforming rows to get zeros under leading coefficients. It's a systematic approach:

• Switch rows if necessary to have a non-zero leading entry
• Scale rows to make leading entries 1
• Eliminate values below the leading entries by suitable row operations

Through these operations, matrices become fractionally easier to digest! Utilize Gaussian elimination to convert awkward numbers into crisp rows where the solutions become more visible – no extra-fancy math tricks required!

Matrix rank

Matrix rank measures how much information a matrix holds. Think of it as telling how many independent rows or columns a matrix has. More simply, it indicates how many distinct types of data it stores without redundancy. To decipher this, consider how row-echelon forms help:

• Count non-zero rows here to determine rank
• If rows talk too much about each other, rank drops

This is what happened with our matrix. While being a 3x3, transforming it revealed only 2 healthy, independent rows. The rank 2 tells us directly: Not every equation contributes new info. Thus, it isn’t full rank (doesn't equal the total number containing rows), and accordingly, no inverse emerges from this data.

Gauss-Jordan elimination

Extending Gaussian elimination, Gauss-Jordan elimination goes the extra mile: achieving not just row-echelon, but reduced row-echelon form. What's that, you ask? Easy—more zeroes! In technical terms, each leading 1 has zeros both above and below it across the matrix. It transforms matrices into identities where possible and unveils inverse matrices if they exist.

The journey goes like this:

• Ensure every leading entry in the rows is 1
• Clear everything above and below this leading 1

Attempting this with matrix A, however, led us to a snag: No full rank. We found zero rows that disrupt forming all 1s instantly, which means no inverse. Strive to achieve Gauss-Jordan results for full rank matrices, paving the way to precise mathematical solutions.