Find (AB)^-1 with Given Matrices | Inverse of 2x2 Matrices

In summary, to find the inverse of a 2x2 matrix, you need to use the formula (AB)^-1 = (B^-1)(A^-1) where A and B are the given matrices. The purpose of finding the inverse of a matrix is to "undo" the original matrix and it is useful in various fields such as solving linear equations and in physics and engineering. Not all matrices can be inverted, as they must be square and have a non-zero determinant. The determinant of a matrix is a real number that can be calculated from the elements of the matrix and is used in various mathematical operations. There is a shortcut for finding the inverse of a 2x2 matrix, which involves using the formula (AB
  • #1

Homework Statement



Use the given matrices to find (AB)^-1

(These are 2 X 2 matrices, please ignore the fraction bar in between the top and bottom elements. I can't figure this stupid latex piece of crap out)

A[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{1}{2}}{\frac{-1}{2}} \ldots \frac{\frac{-5}{2}}{\frac{3}{2}}\right)[/tex]

B[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{2}{3}}{\frac{-1}{3}} \ldots \frac{\frac{4}{3}}{\frac{5}{2}}\right)[/tex]

Homework Equations



Only the one for inverse matrices which states,

(AB)[tex]^{-1}[/tex] = B[tex]^{-1}[/tex]A[tex]^{-1}[/tex]

The Attempt at a Solution



The answer I get in the end is:

(AB)[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{-1}{3}}{\frac{-17}{12}} \ldots \frac{\frac{1}{3}}{\frac{55}{12}}\right)[/tex]

But the book gets,

(AB)[tex]^{-1}[/tex] = [tex]\left(\frac{\frac{-1}{3}}{-1} \ldots \frac{\frac{1}{3}}{\frac{10}{3}}\right)[/tex]

Am I the one doing something wrong, or is the book wrong?

Any help would be greatly appreciated.

Thanks!
 
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  • #2
If you copied the matrices correctly, your answer is correct.
If the 5/2 in (2,2)-entry of [itex]B^{-1}[/itex] is supposed to be 5/3 then the book is correct.

By the way, click the formula to see the LaTeX code:
[tex]
A^{-1} = \begin{pmatrix} \frac{1}{2} & -\frac52 \\ \-\frac{1}{2} & \frac{3}{2} \end{pmatrix} =
\frac12 \begin{pmatrix}
1 & -5 \\
-1 & 3
\end{pmatrix}
[/tex]
 
  • #3
Nope, the book says what I wrote. I'm not surprised though; I've found several typos in the questions in this particular textbook...

Thanks a lot for the clarification, I thought maybe I was missing some obscure rule :p.

Also, thanks for showing how to use the LaTex code properly :)
 
  • #4
Ownaginatious;2156299[h2 said:
Homework Equations[/h2]

Only the one for inverse matrices which states,

(AB)[tex]^{-1}[/tex] = B[tex]^{-1}[/tex]A[tex]^{-1}[/tex]

I find it a lot more easy to calculate AB first,then the inverse...

just a tip,since the main question was already answered
 
  • #5
theallknower said:
I find it a lot more easy to calculate AB first,then the inverse...

just a tip,since the main question was already answered
In this case your approach would take longer, because you first have to calculate A and B from their inverses, then multiply them and finally take the inverse of that, while just multiplying the two given matrices in the correct order gives you the right answer immediately.
 

1. How do I find the inverse of a 2x2 matrix?

To find the inverse of a 2x2 matrix, you can use the following formula:
      (AB)^-1 = (B^-1)(A^-1)
Where A and B are the given matrices. This means you need to find the inverse of each individual matrix, and then multiply them in reverse order.

2. What is the purpose of finding the inverse of a matrix?

The inverse of a matrix is used to "undo" the original matrix. It is especially useful in solving systems of linear equations, finding determinants, and solving problems in physics and engineering.

3. Can all matrices be inverted?

No, not all matrices can be inverted. A matrix must be square (same number of rows and columns) and have a non-zero determinant in order to be inverted.

4. What is the determinant of a matrix?

The determinant of a matrix is a real number that can be calculated from the elements of the matrix. It is used to determine whether a matrix can be inverted, and is also used in various other mathematical operations.

5. Is there a shortcut for finding the inverse of a 2x2 matrix?

Yes, there is a shortcut for finding the inverse of a 2x2 matrix. You can use the following formula:
      (AB)^-1 = 1/det(A)(B11 -B12 -B21 B22)
Where A is the given matrix and det(A) is the determinant of A. This formula can save time in calculations, but it is still important to understand the underlying concepts of matrix inversion.

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