Can you please help with finding an inverse of an elementary matrices?

Yulia
2021-03-07
Answered

Can you please help with finding an inverse of an elementary matrices?

You can still ask an expert for help

lamanocornudaW

Answered 2021-03-08
Author has **85** answers

Step 1

An elementary matrix is a kind of matrices that differ from the identity matrix from a single elementary operation. The elementary operation can be row operation or column operation. Thus after performing one elementary row operation or elementary column operation, the matrix changes to an identity matrix, such type of matrix is called an elementary matrix.

The inverse of elementary matrices is found from the result that when two matrices are multiplied with each other, and the resultant matrix is an identity matrix, then the multiplied matrix is the inverse of the other matrix.

It is represented as,

$A\times {A}^{-1}=I$

Here, A is the elementary matrix,${A}^{-1}$ is the inverse of the elementary matrix and I is the identity matrix.

Step 2

Consider a$(3\times 3)$ elementary matrix A shown below:
$A=\left[\begin{array}{ccc}1& 0& 0\\ -7& 1& 0\\ 0& 0& 1\end{array}\right]$
The above matrix is a elementary matrix since when a column operation is performed the matrix becomes an identity matrix.

The column operation is$({C}_{1}\to {C}_{1}+7\times {C}_{2})$ , In order to find the inverse of the matrix insert a $(3\times 3)$ identity matrix to the left of the elementary matrix and perform the above mentioned column operation to the whole matrix,

Thus, the elementary matrix changes to an identity matrix, and the identity matrix changes to a transformed matrix, and this transformed matrix is the inverse of the elementary matrix.

The steps are shown below:

$A|I=\left[\begin{array}{cccccc}1& 0& 0& 1& 0& 0\\ -7& 1& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 1\end{array}\right]$

$({C}_{1}\to {C}_{1}+7\times {C}_{2})$

$A|I=\left[\begin{array}{cccccc}1& 0& 0& 1& 0& 0\\ 0& 1& 0& 7& 1& 0\\ 0& 0& 1& 0& 0& 1\end{array}\right]$
Thus, the transformed matrix is,

${A}^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ 7& 1& 0\\ 0& 0& 1\end{array}\right]$
This is possible since,

$A\times {A}^{-1}=I$

$\left[\begin{array}{ccc}1& 0& 0\\ -7& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 7& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

An elementary matrix is a kind of matrices that differ from the identity matrix from a single elementary operation. The elementary operation can be row operation or column operation. Thus after performing one elementary row operation or elementary column operation, the matrix changes to an identity matrix, such type of matrix is called an elementary matrix.

The inverse of elementary matrices is found from the result that when two matrices are multiplied with each other, and the resultant matrix is an identity matrix, then the multiplied matrix is the inverse of the other matrix.

It is represented as,

Here, A is the elementary matrix,

Step 2

Consider a

The column operation is

Thus, the elementary matrix changes to an identity matrix, and the identity matrix changes to a transformed matrix, and this transformed matrix is the inverse of the elementary matrix.

The steps are shown below:

Jeffrey Jordon

Answered 2022-01-22
Author has **2262** answers

Answer is given below (on video)

asked 2021-01-31

Find a basis for the space of $2\times 2$ diagonal matrices.

$\text{Basis}=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

asked 2021-02-08

Let B be a 4x4 matrix to which we apply the following operations:

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

1. double column 1,

2. halve row 3,

3. add row 3 to row 1,

4. interchange columns 1 and 4,

5. subtract row 2 from each of the other rows,

6. replace column 4 by column 3,

7. delete column 1 (column dimension is reduced by 1).

(a) Write the result as a product of eight matrices.

(b) Write it again as a product of ABC (same B) of three matrices.

asked 2022-06-14

How to Find the Differential of $y=2{\mathrm{sin}}^{2}(x)$ when $x=\pi /4$ and $dx=0.49$

asked 2022-04-09

How to prove

$\mathrm{tan}x+\mathrm{tan}(x+\frac{\pi}{3})+\mathrm{tan}(x+\frac{2\pi}{3})=3\mathrm{tan}3x$

asked 2022-05-06

What is (3,-4) plus (2,-4)?

asked 2022-04-20

Variation of $x\to \mathrm{tan}x$ over $(\frac{\pi}{2},\pi ]$ without using derivation?

asked 2022-03-01

-x-+y=1 -3x+y=7What is the solution