1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
NemiM [27]
2 years ago
13

How to do the inverse of a 3x3 matrix gaussian elimination.

Mathematics
1 answer:
nata0808 [166]2 years ago
6 0

As an example, let's invert the matrix

\begin{bmatrix}-3&2&1\\2&1&1\\1&1&1\end{bmatrix}

We construct the augmented matrix,

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 2 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right]

On this augmented matrix, we perform row operations in such a way as to transform the matrix on the left side into the identity matrix, and the matrix on the right will be the inverse that we want to find.

Now we can carry out Gaussian elimination.

• Eliminate the column 1 entry in row 2.

Combine 2 times row 1 with 3 times row 2 :

2 (-3, 2, 1, 1, 0, 0) + 3 (2, 1, 1, 0, 1, 0)

= (-6, 4, 2, 2, 0, 0) + (6, 3, 3, 0, 3, 0)

= (0, 7, 5, 2, 3, 0)

which changes the augmented matrix to

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 7 & 5 & 2 & 3 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 \end{array} \right]

• Eliminate the column 1 entry in row 3.

Using the new aug. matrix, combine row 1 and 3 times row 3 :

(-3, 2, 1, 1, 0, 0) + 3 (1, 1, 1, 0, 0, 1)

= (-3, 2, 1, 1, 0, 0) + (3, 3, 3, 0, 0, 3)

= (0, 5, 4, 1, 0, 3)

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 7 & 5 & 2 & 3 & 0 \\ 0 & 5 & 4 & 1 & 0 & 3 \end{array} \right]

• Eliminate the column 2 entry in row 3.

Combine -5 times row 2 and 7 times row 3 :

-5 (0, 7, 5, 2, 3, 0) + 7 (0, 5, 4, 1, 0, 3)

= (0, -35, -25, -10, -15, 0) + (0, 35, 28, 7, 0, 21)

= (0, 0, 3, -3, -15, 21)

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 7 & 5 & 2 & 3 & 0 \\ 0 & 0 & 3 & -3 & -15 & 21 \end{array} \right]

• Multiply row 3 by 1/3 :

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 7 & 5 & 2 & 3 & 0 \\ 0 & 0 & 1 & -1 & -5 & 7 \end{array} \right]

• Eliminate the column 3 entry in row 2.

Combine row 2 and -5 times row 3 :

(0, 7, 5, 2, 3, 0) - 5 (0, 0, 1, -1, -5, 7)

= (0, 7, 5, 2, 3, 0) + (0, 0, -5, 5, 25, -35)

= (0, 7, 0, 7, 28, -35)

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 7 & 0 & 7 & 28 & -35 \\ 0 & 0 & 1 & -1 & -5 & 7 \end{array} \right]

• Multiply row 2 by 1/7 :

\left[ \begin{array}{ccc|ccc} -3 & 2 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 4 & -5 \\ 0 & 0 & 1 & -1 & -5 & 7 \end{array} \right]

• Eliminate the column 2 and 3 entries in row 1.

Combine row 1, -2 times row 2, and -1 times row 3 :

(-3, 2, 1, 1, 0, 0) - 2 (0, 1, 0, 1, 4, -5) - (0, 0, 1, -1, -5, 7)

= (-3, 2, 1, 1, 0, 0) + (0, -2, 0, -2, -8, 10) + (0, 0, -1, 1, 5, -7)

= (-3, 0, 0, 0, -3, 3)

\left[ \begin{array}{ccc|ccc} -3 & 0 & 0 & 0 & -3 & 3 \\ 0 & 1 & 0 & 1 & 4 & -5 \\ 0 & 0 & 1 & -1 & -5 & 7 \end{array} \right]

• Multiply row 1 by -1/3 :

\left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & 1 & -1 \\ 0 & 1 & 0 & 1 & 4 & -5 \\ 0 & 0 & 1 & -1 & -5 & 7 \end{array} \right]

So, the inverse of our matrix is

\begin{bmatrix}-3&2&1\\2&1&1\\1&1&1\end{bmatrix}^{-1} = \begin{bmatrix}0&1&-1\\1&4&-5\\-1&-5&7\end{bmatrix}

You might be interested in
Which 2 expressions are equiavalent to -40/9
Sidana [21]

Answer:

360 is the answer

Step-by-step explanation:

hi

have a good day

hope it helps you

thank me later

carryonlearing

8 0
2 years ago
Read 2 more answers
In math class the girl to boy ratio is 8 to 6 if there are 24 girls in the class , how many boys are there ?
Volgvan

Answer:  The required number of boys in the class is 18.

Step-by-step explanation: Given that in math class, the girl to boy ratio is 8 to 6 and there are 24 girls in the class.

We are to find the number of boys in the class.

Let 8x and 6x represents the number of girls and boys in the class.

Then, according to the given information, we have

8x=24\\\\\Rightarrow x=\dfrac{24}{8}\\\\\Rightarrow x=3.

Therefore, the number of boys in the class is given by

6x=6\times3=18.

Thus, the required number of boys in the class is 18.

7 0
3 years ago
What is the domain and range , and is it a function ?
dem82 [27]

Answer:

The domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes. (In grammar school, you probably called the domain the replacement set and the range the solution set.

6 0
3 years ago
Deshun went to visit zakia and he brought 9 video games, which is 10% of his collection. using the number line above, determine
Elena-2011 [213]
10% of what is 9...
0.10x = 9
x = 9/0.10
x = 90.....he has 90 all together
4 0
3 years ago
Write four and twenty-six thousandths in standard form
juin [17]

426,000 * 10^-4. You can go use a standard form calculator. and if you don't have a physical one, you can use one online.

6 0
3 years ago
Other questions:
  • Lines g, h, and l are parallel and m ∠ 1 = 52°. <br><br> What is m ∠ 11?
    5·2 answers
  • Im giving free brainlest only if you give it to me
    12·2 answers
  • Can someone please help?
    15·2 answers
  • 3
    11·1 answer
  • A researcher drilled a hole in a die and filled it with a lead​ weight, then proceeded to roll it 209 times. The observed freque
    8·1 answer
  • Which monomial is a perfect cube?
    8·2 answers
  • Which statement is correct when a scatter plot is constructed
    5·1 answer
  • An artist buys 36 tubes of oil paint, 6 paintbrushes, 3 canvases, and 2 jars of paint thinner. What is the ratio of paintbrushes
    7·1 answer
  • If the expression (2x⁴)³ was written in axb form, which of the following would be the sum of a and b?
    10·2 answers
  • WILL GIVE BRAINLIEST What is the slope of the table shown? <br> 3<br> 5<br> 1<br> 2
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!