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As-12

Inversing Matrices

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I am studying about linear algebra , Inversing matrix and I dont understand it at all. The book said about Jordan's medthod to inversing but I dont get it. And To det matrices det[ a b ] [ c d ] = a*d - b*c right? If in 3 dimensions ? How we will det it? Sry but its pretty hard to me bcoz Im only in grade 8.

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Quote:
Original post by As-12
I am studying about linear algebra , Inversing matrix and I dont understand it at all. The book said about Jordan's medthod to inversing but I dont get it.
And To det matrices
det[ a b ]
[ c d ] = a*d - b*c right?
If in 3 dimensions ? How we will det it?
Sry but its pretty hard to me bcoz Im only in grade 8.
Matrix inversion is actually a pretty difficult topic, so don't expect to understand it all at once (I don't claim to understand it myself, for that matter). Just take your time with it - you're only in 8th grade, so you've got plenty of it :-)

Matrix inversion is closely related to systems of linear equations, so that's where you'll want to start. Here's a simple system of 2 equations in 2 unknowns:
ax+by = e
cx+dy = f
There are various ways to go about solving such a system. One way is to start by solving for y in terms of x:
y = (f-cx)/d
Plug this into the first equation to get one equation in one unknown:
ax+b(f-cx)/d = e
Do some algebra to solve for x:
x = (de-bf)/(ad-bc)
Plug x into the equation for y and solve:
y = (-ce+af)/(ad-bc)
The solutions are then:
x = ( de-bf)/(ad-bc)
y = (-ce+af)/(ad-bc)
I'm not going to write it all out here, but if you express the system and its solution in matrix form, you'll begin to see the motivation for matrix inversion; the original coefficient matrix is:
[a b]
[c d]
While the inverse is:
[  d(ad-bc) -b(ad-bc) ]
[ -c(ad-bc) a(ad-bc) ]
Note that the denominator of each element is the determinant of the matrix, which offers further clues as to the significance of that particular operation.

I don't know how helpful that was, but maybe it will at least get you moving in the right direction.

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Quote:
Original post by As-12
I am studying about linear algebra , Inversing matrix and I dont understand it at all. The book said about Jordan's medthod to inversing but I dont get it.
And To det matrices
det[ a b ]
[ c d ] = a*d - b*c right?
If in 3 dimensions ? How we will det it?
Sry but its pretty hard to me bcoz Im only in grade 8.


Consider the following operations you can perform on the rows (or columns) of a matrix:
1st) Exchange two of them.
2nd) Scale the numbers in a row by a constant number.
3rd) Add to a row, another one, scaled by a constant number.

The first one negates the sign of the determinant each time it is performed, the second scales the determinant by the very same number, and the third leaves it unaffected.
Using these operations, it is possible to turn any square matrix into an identity matrix. It turns out though, that if you replicate the very same operations on the identity matrix, the result will be the inverse of the original.
This procedure is also widely used in finding the determinant of a matrix by turning it into a triangular matrix (one whose all elements below or above the major diagonal are zero) because then its determinant is simply the product of the numbers in the major diagonal.
The initial matrix and the triangular one will have the same determinant because only operation (3) is used, and it does not affect the determinant of a matrix.

Determinants of 3x3 (and higher) matrices are found in a recursive manner.
If you have the matrix

[ 1 5 7 ]
[-2 2 1 ]
[ 3 0 2 ]

choose a row or a column, e.g. the second row. Then, the determinant is equal to the sum of all numbers in that row, each one multiplied by the determinant that arises if you exclude the column and row of that number from the matrix, and also multiplied by -1 or +1 according to the position of that number, with the following rule

[ + - + ]
[ - + - ]
[ + - + ]

E.g. the determinant of the previous matrix, (using the 2nd row) would be:

D = (-1)*(-2)*|5 7| + (+1)*(2)*|1 7| + (-1)*(1)*|1 5| =
|0 2| |3 2| |3 0|

= 2*(10 - 0) + 2*(2-21) - (-15) = 20 -38 + 15 = -3

You will get the same result no matter what row or column you choose.
If it was a 4x4 or larger matrix, it's exactly the same thing. You break all determinants that will show up, down to 2x2, using the very same rule.

I hope this helps. If you still don't get something, just ask, however you should know that there is quite a lot of theory behind these.

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