# Linear Algerbra. Inverses and Algerbraic Properties of Matrices

• stumpoman
In summary: Left-multiply the third equation with B.In summary, this student is trying to solve for D in homework, but does not know where to start. They first try to do what is shown in the text, but then realize that it may not be the right way to do it. They go on to try and solve for D by left-multiply ing the third equation with B.
stumpoman

## Homework Statement

Assuming that all matrices, A, B, C, and D, are n x n and invertible, solve for D.

$C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}$

## Homework Equations

$C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}$

## The Attempt at a Solution

I must have missed something in the reading of this section. All I can think of is

$B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=I$

but I don't know where to go from there or if it is even the right way to start.

Left-multiply the third equation with B.

stumpoman said:

## Homework Statement

Assuming that all matrices, A, B, C, and D, are n x n and invertible, solve for D.

$C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}$
Just "undo" every thing that is done to D, one step at a time. For example if you multiply both sides, on the left, by $(C^T)^{-1}$ you get $B^{-1}A^{-2}BAC^{-1}DA^{-2}B^TC^{-2}= (C^T)^{-1}C^T= I$
Then multiply on the left by $B$, then on the right by $C^2$, or do both together to get $A^{=2}BAC^{-1}DA^{-2}B^T= BC^2$.
Continue to "unpeel" D.

## Homework Equations

$C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T}$

## The Attempt at a Solution

I must have missed something in the reading of this section. All I can think of is

$B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=I$

but I don't know where to go from there or if it is even the right way to start.
Yes, that was the first step as I showed. Continue in the same way.

## 1. What is a matrix and how is it used in linear algebra?

A matrix is a rectangular array of numbers or variables arranged in rows and columns. In linear algebra, matrices are used to represent systems of linear equations and perform operations like addition, subtraction, and multiplication.

## 2. What is the inverse of a matrix and why is it important?

The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix. It is important because it allows us to solve systems of equations and perform other operations that would otherwise be impossible.

## 3. How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or the adjoint method. Both methods involve performing a series of row operations to transform the matrix into its reduced row-echelon form.

## 4. What are the algebraic properties of matrices?

The algebraic properties of matrices include commutativity, associativity, and distributivity. This means that matrix addition and multiplication follow the same rules as regular addition and multiplication.

## 5. How are matrices used in real-world applications?

Matrices are used in a variety of real-world applications, such as computer graphics, engineering, economics, and statistics. They are used to represent and solve systems of equations, analyze data, and make predictions.

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