How do you find the adjoint of a 3x3 matrix? can u explain example?

In summary, the conversation discusses the concept of finding the adjoint of a nxn matrix. The process involves deleting a row and column from the original matrix and taking the determinant, then multiplying by (-1)^{i+j}. A specific example is given for a 3x3 matrix.
  • #1
mr_coffee
1,629
1
Hello everyone, I think i don't understand the inverses because i don't understand how u find the adjoint of a nxn matrix. The book has this example and i have no idea how they got from A to A adj, makes no sense to me!

Here is the picture:
http://img89.imageshack.us/img89/3010/lastscan0oo.jpg
if that link is slow try:L
http://show.imagehosting.us/show/806170/0/nouser_806/T0_-1_806170.jpg
thanks! :biggrin:
 
Last edited by a moderator:
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  • #2
mr_coffee said:
Hello everyone, I think i don't understand the inverses because i don't understand how u find the adjoint of a nxn matrix. The book has this example and i have no idea how they got from A to A adj, makes no sense to me!
Here is the picture:
http://img89.imageshack.us/img89/3010/lastscan0oo.jpg
if that link is slow try:L
http://show.imagehosting.us/show/806170/0/nouser_806/T0_-1_806170.jpg
thanks! :biggrin:

To find the adjoint of a A:

[tex]
adj(A)_{ij} = (-1)^{i+j} det[A(j|i)]
[/tex]

That means that the entry in the row [tex]i[/tex] and column [tex]j[/tex] of [tex]adj(A)[/tex] is obtained by deleting the column [tex]j[/tex] and the row [tex]i[/tex] of A and then taking the determinant of that and multipliying by [tex](-1)^{i+j}[/tex]

For instance if u have a 3x3 matrix:
http://en.wikipedia.org/math/e36e0138b126ebbcf8fe80cd4f58f3aa.png

this is the adjoint:
http://en.wikipedia.org/math/a3c81ad8680add569a7377cda2529147.png
 
Last edited by a moderator:
  • #3
Ahh thank u so much!
 

1. What is an adjoint matrix?

An adjoint matrix is a square matrix that is created by taking the transpose of the cofactor matrix of the original matrix. It is also known as the adjugate matrix or classical adjoint.

2. Why do we need to find the adjoint of a matrix?

The adjoint matrix is used to calculate the inverse of a matrix, which is a useful tool in solving linear equations and performing other operations on matrices.

3. How do you find the adjoint of a 3x3 matrix?

To find the adjoint of a 3x3 matrix, you first need to calculate the cofactor matrix by finding the determinant of each minor matrix. Then, take the transpose of the cofactor matrix to get the adjoint matrix.

4. Can you provide an example of finding the adjoint of a 3x3 matrix?

Let's say we have the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. To find the adjoint, we first find the determinant of each minor matrix. For example, the minor matrix for the element 1 is [[5, 6], [8, 9]], so the determinant is (5*9 - 6*8) = -3. Then, we create the cofactor matrix by placing the determinants in their corresponding positions, with alternating signs. The cofactor matrix for A would be: [[-3, 3, -3], [3, -6, 3], [-3, 3, -3]]. Finally, we take the transpose of the cofactor matrix to get the adjoint matrix: [[-3, 3, -3], [3, -6, 3], [-3, 3, -3]].

5. What is the relationship between the adjoint and the inverse of a matrix?

The adjoint matrix is used to find the inverse of a matrix. The inverse of a matrix A is equal to the adjoint of A divided by the determinant of A. This relationship is expressed as A^-1 = adj(A) / det(A).

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