# Solving AB=0 for B given A Matrix

• Dell
In summary, the conversation discusses the possible values of matrix B given a specific matrix A and the fact that their product equals to the zero matrix. The concept of nullspace is mentioned as a helpful tool in solving this problem, and it is noted that the columns of B must be different multiples of (-2, 1, 1) in order for AB to equal the zero matrix. The conversation also mentions using parameters to represent the columns of B and checking this solution by calculating AB.
Dell
gven a matrix A=

1 0 2
1 1 1
5 2 8

and knowing AB=0 ,B$$\neq$$0
what are possible values of B

is there any way to solve this other than
making B a matrix of parameters , doing the multiplication and solving, ie
B=

x y z
a b c
d e f

x+0a+2d=0
y+0b+2e=o
... etc

It's helpful to know about the nullspace of a matrix in this problem. In general, the nullspace is the set of vectors x such that Ax = 0.

For this problem, the nullspace is one-dimensional, and consists of all scalar multiples of (-2, 1, 1).

Instead of looking at AB = 0, think about what's happening to the individual columns of B, call them B_1, B_2, and B_3. What can you say about A*B_1 = 0? A*B_2 = 0? A*B_3 = 0?

all got to be multiples of (-2 1 1)? am i on the right track, haven't yet learned about nullspace.

Yes and yes, so congratulations! Keep in mind that the columns are different multiples of (-2, 1, 1). (Hint: use parameters.)

To check, write a matrix B as above and calculate AB. Should come out with the 3 x 3 zero matrix.

## 1. What does solving for B given A matrix mean?

Solving for B given A matrix means finding the values of the matrix B that satisfy the equation AB=0, where A is a given matrix.

## 2. Why is solving AB=0 for B given A matrix important?

This type of problem often arises in linear algebra and is important for understanding matrix operations and solving systems of linear equations.

## 3. What are the steps for solving AB=0 for B given A matrix?

The steps for solving this problem involve using matrix operations, such as row reduction, to manipulate the given matrix A until it is in reduced row-echelon form. Then, using the properties of matrix multiplication, the values of B can be determined.

## 4. Can you provide an example of solving AB=0 for B given A matrix?

Sure, let's say the matrix A is given as [1 2 3; 4 5 6; 7 8 9]. We can manipulate this matrix using row reduction to obtain [1 0 -1; 0 1 2; 0 0 0]. This means that B can be any matrix of the form [x; -2x; x], where x is any scalar value.

## 5. What are some applications of solving AB=0 for B given A matrix?

This type of problem is commonly used in fields such as engineering, physics, and computer science for solving systems of linear equations and performing transformations on vectors and matrices. It is also used in machine learning and data analysis.

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