# Find a 3x3 Matrix such that....

• Jtechguy21
In summary, the matrix A does not have a range of output values when multiplied by any column vector.
Jtechguy21

## Homework Statement

Find a non zero matrix(3x3) that does not have
in its range. Make sure your matrix does as it should.

## The Attempt at a Solution

[/B]
I know a range is a set of output vectors, Can anyone help me clarify the question?
I'm just not sure specifically what its asking of me, in regards to the matrix I am looking for.

You are asked to find a 3 by 3 matrix, A, such that Ax is NOT equal to $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ for any x. This does not have a single correct aswer. The trivial solution, the zero matrix, would be such a matrix because the zero matrix times any x is the 0 vector- but that is specifically excluded- you must give a "non-zero" matrix. There are still an infinite number of such matrices. What about a matrix that makes the first component of any vector 0? Or the second or third component? (But not all three).

Jtechguy21 said:

## Homework Statement

Find a non zero matrix(3x3) that does not have
in its range. Make sure your matrix does as it should.

## The Attempt at a Solution

[/B]
I know a range is a set of output vectors, Can anyone help me clarify the question?
I'm just not sure specifically what its asking of me, in regards to the matrix I am looking for.

You are being asked to find a ##3 \times 3## matrix ##A## for which the vector ##\vec{r} = \langle 1,2,3 \rangle^T ## (transpose) is an impossible output; that is, you want##A \vec{x} \neq \vec{r}## for ALL input vectors ##\vec{x} \in R^3##.

We haven't learned about tranpose yet in class(but i just briefly looked it up right now)

What is the purpose of taking the transpose of the the column vector 1 2 3 in this case?

So I find this A= 3x3 matrix that when I multiply it by any column vector(x,y,z) the ouput does not equal r<1,2,3>T is this correct?

Jtechguy21 said:
We haven't learned about tranpose yet in class(but i just briefly looked it up right now)

What is the purpose of taking the transpose of the the column vector 1 2 3 in this case?

So I find this A= 3x3 matrix that when I multiply it by any column vector(x,y,z) the ouput does not equal r<1,2,3>T is this correct?

Is that not what I just said?

Anyway, writing a transpose simplifies typing. Instead of taking 3 lines to write the column vector like this:
$$\left[ \begin{array}{c}1\\2\\3 \end{array} \right]$$
we can write it in one line, like this: ##(1,2,3)^T## or ##\langle 1,2,3 \rangle^T##, as the transpose of a row-vector.

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You can use rank-nullity: just use a matrix that "kills" the vector ## (1,2,3)^T ##, say.

Ray Vickson said:
Is that not what I just said?

Anyway, writing a transpose simplifies typing. Instead of taking 3 lines to write the column vector like this:
$$\left[ \begin{array}{c}1\\2\\3 \end{array} \right]$$
we can write it in one line, like this: ##(1,2,3)^T## or ##\langle 1,2,3 \rangle^T##, as the transpose of a row-vector.

Thanks for explaining that to me.

WWGD said:
You can use rank-nullity: just use a matrix that "kills" the vector ## (1,2,3)^T ##, say.

Sounds interesting. Unfortunately we have not covered this yet(2nd week in) but I am watching a youtube videos to see how this "Killing" things works.

HallsofIvy said:
You are asked to find a 3 by 3 matrix, A, such that Ax is NOT equal to $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ for any x. This does not have a single correct aswer. The trivial solution, the zero matrix, would be such a matrix because the zero matrix times any x is the 0 vector- but that is specifically excluded- you must give a "non-zero" matrix. There are still an infinite number of such matrices. What about a matrix that makes the first component of any vector 0? Or the second or third component? (But not all three).

thanks for your input. I had a feeling that's exactly what it meant, but sometimes i doubt my self with these kind of things. and when I was trying to understand the problem I figured there was more than one answer(you confirmed it)

When you refer to x, does that mean the column vector with x,y,z (any x ,y,z, values) which when multiplied by Matrix A does not equal specified range?

Jtechguy21 said:
Sounds interesting. Unfortunately we have not covered this yet(2nd week in) but I am watching a youtube videos to see how this "Killing" things works.

I don't know if you have had the following material yet, but in 2d or 3d it is "obvious" from diagrams: given a fixed vector ##\vec{u}##, any vector ##\vec{v}## can be decomposed into a component ##\vec{v}_{||}## parallel to ##\vec{u}## and a component ##\vec{v}_{\perp}## perpendicular to ##\vec{u}##. (Think of decomposing a force in Physics into components parallel and perpendicular to some given direction.)

What would happen if, for every vector ##\vec{v}## you took the perpendicular component ##\vec{v}_{\perp}##? Is the operation ##\vec{v} \rightarrow \vec{v}_{\perp}## a linear operation?

My idea is this: you can use a matrix whose kernel is either 1- , 2- , or 3- dimensional ( if the kernel is 0-dimensional, the matrix is invertible, so ## Ax=b ## has a solution for any b, including, in particular, ##(1,2,3)^T##. If the kernel is 3-dimensional, you get the ##0##-matrix ). So include , in each case, ##(1,2,3)^T## as a basis vector in the kernel of a matrix. You can also use the fundamental theorem of linear algebra:

https://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra

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## 1. How do I find a 3x3 matrix with specific elements?

To find a 3x3 matrix with specific elements, first determine the values you want in each position of the matrix. Then, use the formula aij = ni * mj (where aij represents the value in the i-th row and j-th column, ni represents the value in the i-th position in the first row, and mj represents the value in the j-th position in the first column) to fill in the remaining values.

## 2. Can I use any numbers in the 3x3 matrix?

Yes, you can use any real numbers in a 3x3 matrix. However, in some applications, there may be restrictions on the type of numbers that can be used, such as only using integers or only using positive numbers.

## 3. What is the purpose of finding a 3x3 matrix?

3x3 matrices are commonly used in linear algebra to represent systems of equations, transformations, and other mathematical operations. They are also used in computer graphics, physics, and other fields to represent data and perform calculations.

## 4. How many different 3x3 matrices are there?

There are an infinite number of possible 3x3 matrices, as there are infinite combinations of real numbers that can be used as elements. However, in some cases, there may be a limited number of matrices that meet specific criteria or have certain properties.

## 5. How do I know if a 3x3 matrix is invertible?

A 3x3 matrix is invertible if its determinant (ad-bc) is not equal to 0. If the determinant is 0, the matrix is singular and does not have an inverse. Additionally, a 3x3 matrix is invertible if its rank is equal to 3, meaning all of its rows and columns are linearly independent.

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