Finding inverse matrices using Guass approach

• thomas49th
In summary: The only thing that might have saved you some time would have been to recognize that the bottom row of I was not correct, because the bottom row of A was not all 0's.
thomas49th

Homework Statement

Use the determinate method and also the Guass elimination method to find the inverse of the following matrix. Check your results by direct multiplication
$$A =\left | \begin{array}{ccc} 2&1&0\\ 1&0&0\\ 4&1&2 \end{array}\right | =$$

Let's do Guass first

Homework Equations

Place A by I and attemp to get A into I. Everything I perform on A must be performed on I and when A is in I, the original I is the inverse?

$$A =\left | \begin{array}{ccc} 2&1&0\\ 1&0&0\\ 4&1&2 \end{array}\right |$$
$$I=\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right |$$

The Attempt at a Solution

Interchange rows 1 and R
R1 <-> R2

$$A =\left | \begin{array}{ccc} 1&0&0\\ 2&1&0\\ 4&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array}\right |$$

Now R2 - 2R1

$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 4&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ 0&0&1 \end{array}\right |$$

R3 - 4R1
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&1&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ 0&-4&1 \end{array}\right |$$
R3 - R2
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&2 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ -1&-4&1 \end{array}\right |$$

R3 / 2
$$A =\left | \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right | =$$
$$I =\left | \begin{array}{ccc} 0&1&0\\ 1&-2&0\\ -1/2&-2&1/2 \end{array}\right |$$

but multiplying my A and new I together does not give me I? What have I done wrong?

Check the R3-R2 step. (-4)-(-2)=(-2) not (-4).

You made a mistake in the calculation right after what you show as R3 - R2.
The -4 in the last row of I should be -2.

ahh got it working by starting again and doing another combination... made a silly mistake in this jumble!

Thanks for the pointers guys. My other route still took just as long as this one. Do any of you any tricks to improve Guass Elimination efficiency or is it just experience.

Thanks
Tom

I don't know how you are doing the calculations, but your work was difficult to check with two separate matrices. The usual practice is to create an augmented matrix, starting with A in the left half and I in the right half. After you get to I in the left half, your inverse of A will be in the right half. Each augmented matrix will look something like this.

$$\left[ \begin{array}{ccccccc} 2&1&0&|&1&0&0\\ 1&0&0&|&0&1&0\\ 4&1&2&|&0&0&1 \end{array}\right ]$$

An "accounting" tip to make it easier to understand what you did is to use a notation that indicates which row changes when you add a multiple of one row to another. One way to do this is fairly verbose R1 <-- R1 - 3R2. You can abbreviate this to R1 - 3R2 if it's understood that it's always the first row listed that is added to. Obviously if you just switch two rows, it doesn't matter which one you list first, and if you replace a row by a multiple of itself, there's only one row involved, so there shouldn't be any confusion about which rows are involved.

Other than that, you row-reduced your matrices the way I would have, so I don't see anything that you could have done that would have economized your efforts.

What is the Gauss approach for finding inverse matrices?

The Gauss approach, also known as Gaussian elimination, is a method for finding the inverse of a square matrix by performing a series of row operations to transform the matrix into reduced row echelon form.

Why is finding inverse matrices important in scientific research?

Finding inverse matrices is important in scientific research because it allows for the efficient solving of systems of linear equations and can be applied to various mathematical models and simulations.

What are the steps involved in using the Gauss approach to find inverse matrices?

The steps involved in using the Gauss approach to find inverse matrices are:1. Augment the given matrix with the identity matrix2. Use row operations to transform the given matrix into reduced row echelon form3. If the resulting matrix is the identity matrix, then the original matrix is invertible and the transformed matrix is its inverse4. If the resulting matrix is not the identity matrix, then the original matrix is not invertible

What are the limitations of using the Gauss approach to find inverse matrices?

The Gauss approach may not work for matrices that are not square or for matrices that are singular (have a determinant of 0). It also involves a significant amount of computation, especially for larger matrices, which can make it time-consuming.

Are there any alternative methods for finding inverse matrices?

Yes, there are other methods for finding inverse matrices such as using the adjugate matrix, the Cramer's rule, or the LU decomposition method. Each method may have its own advantages and limitations, and the most suitable method may depend on the specific characteristics of the given matrix.

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