It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I was just wondering that because one can multiply and add and subtract matrices, why can't one divide them?
You CAN divide by matrices. To understand what division in the context of matrices mean, let us look at what division means in the context of real numbers. However, in matrix algebra, multiplication is not commutative.
Hence, you need to specify, whether you are dividing by a matrix on the right or on the left. Also, just like division by zero is not possible in the context of real numbers, you cannot divide by certain matrices, which are called singular matrices.
You sort of can. Not all nonzero matrices are invertible. Matrices do not commute, i. Further, with nonzero numbers, there is a unique way to do this. With matrices, a lot of this breaks down. For some matrices, an inverse matrix both exists and is unique, and only for those matrices is it OK to do division in the way you are trying to do it.
Sometimes you can, sometimes you can't. Proposition 1. Let F be a function from to satisfying. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere [4].
Therefore, the division by zero will give great impacts to complex analysis and to our ideas for the space and universe. However, the division by zero 2 is now clear; indeed, for the introduction of 2 , we have several independent approaches as in:. Furthermore, in [5] we gave the results in order to show the reality of the division by zero in our world:. D by considering rotation of a right circular cone having some very interesting phenomenon from some practical and physical problem.
Meanwhile, J. Barukcic and I. Furthermore, T. Reis and J. Anderson [8] [9] extend the system of the real numbers containing division by zero. For the definitions and 2 , we should note that they are not the usual fractions defined by and as the inverses of products that mean contradictions, immediately.
They are just given as definitions for the pairs and , respectively. This precise meaning is given by Proposition 1. Note that for the introduction of the Y-field [5] containing the division by zero, the meaning is the same. For calculations containing the division by zero, we can apply the Y-field laws. In particular, note that the general product property.
In this paper, we will discuss the division by zero in matrices and we will be able to see that the division by zero is our elementary and fundamental mathematics. We will introduce a new space for the Euclidean plane. Indeed, for the point at infinity on the Riemann sphere, we will introduce a new idea and fact. By the division by zero, we can understand that if , then the common point is always given by. The division by zero, in particular, means, surprisingly, that the point at infinity is represented by zero, that is, the coincidence of the point at infinity and the origin.
Precisely, the point at infinity topological point is represented by with the number. The point at infinity is a point of one-point compactification of Aleksandrov and is not represented by the number of the infinity as in the common sense.
We can see that the whole line on the plane passes the point at infinity, by the stereographic projection into the Riemann sphere. The point at infinity is represented by the zero and so, every line on the plane passes the origin in this sense. This fact may be understood that the point at infinity is reflected to the origin. In this sense, the origin will have double natures of the native origin and reflection of the point at infinity.
The latter has a strong discontinuity. Recall the definition of by e-d logic; that is, if and only if for any large , there exists a number such that for any z satisfying ,.
The behavior of the space around the point at infinity may be considered by that around the origin by the linear transform [4]. We thus see that. Let us look at some examples so that we can have a clear idea about them. Until now, you have learned how to add two things in matrices such as variables, numbers, equations amongst others.
But addition does not always work with matrices. Although matrices are added with every entry, we need to add two numbers like 2 and 2, 1 and 8, then 3 and 4, 4 and 6. But what else can we do when adding the numbers 6 and 7 and which have no straight numbers in the other matrix? So, the answer is —. This is always the case when adding the matrices, you need both the matrices of the same dimensions. If they are not of equal sizes, then the addition is not applicable.
It does not make any mathematical logic for adding the nonequal matrices. Subtraction also works with every entry and with same conditions applied. This is the case for both addition and subtraction of matrices. Thus, with the equality of matrix works with entry wise, we compare these entries for creating the simple equations that we can solve. This is nonzero, so it is possible to find the inverse. Find the determinant of a larger matrix.
If your matrix is 3 x 3 or larger, finding the determinant takes a bit more work: 3 x 3 matrix : Choose any element and cross out the row and column it belongs to. Find the determinant of the remaining 2 x 2 matrix, multiply by the chosen element, and refer to a matrix sign chart to determine the sign. Repeat this for the other two elements in the same row or column as the first one you chose, then sum all three determinants.
Read this article for step-by-step instructions and tips to speed this up. Larger matrices : Using a graphing calculator or software is recommended. The method is similar to the 3 x 3 matrix method, but is tedious by hand. Continue on. If your matrix is not square, or if its determinant is zero, write "no unique solution. If the matrix is square and its determinant is non-zero, continue to the next section for the next step: finding the inverse. Part 2. Switch the positions of the elements on the main 2 x 2 diagonal.
If your matrix is 2 x 2, you can use a shortcut to make this calculation much easier. If you'd like to calculate it by hand, refer to the end of this section. Take the opposite of the other two elements, but leave them in position.
Take the reciprocal of the determinant. You found the determinant of this matrix in the section above, so there's no need to calculate it a second time. Multiply the new matrix by the reciprocal of the determinant. Multiply each element of the new matrix by the reciprocal you just found.
Confirm the inverse is correct. To check your work, multiply the inverse by the original matrix. Here's a refresher on how to multiply matrices. Note: Matrix multiplication is not commutative: the order of the factors matters. However, when multiplying a matrix by its inverse, both options will result in the identity matrix.
Review matrix inversion for 3 x 3 matrices or larger. Unless you are learning this process for the first time, save yourself time by using a graphing calculator or math software for larger matrices. If you do need to calculate it by hand, here's a quick summary of one method: [9] X Research source [10] X Research source Adjoin the identity matrix I to the right side of your matrix.
The identity matrix has "1" elements along the main diagonal, and "0" elements in all other positions. Perform row operations to reduce the matrix until the left side is in row-echelon form, then continue reducing until the left side is the identity matrix.
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