Finding Inverse of A Matrix: Example A

In summary, to find the inverse of a matrix, you can write the identity matrix next to the original matrix and perform row operations until the left half becomes the identity matrix. This will result in the inverse matrix on the right. Row operations include addition, substitution, and multiplication by a scalar. It is important to also apply the same operations to the identity matrix.
  • #1
mrroboto
35
0
How do you take the inverse of a matrix?

The specific example I have is

A=

1 1 1 1
1 1 1 3
1 1 3 3
1 3 3 3

Find A^-1
 
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  • #2
There are loads of ways. The quickest is probably to write the identity matrix to the right of A, and then to perform row operations such that the left hald becomes the identity. The matrix on the right is then the inverse.
 
  • #3
what do you mean by "row operations?

so I take

1 1 1 1
1 1 1 3
1 1 3 3
1 3 3 3

and put

1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1

next to it...and then what? add? multiply?
 
  • #4
Is this a homework problem? Are you in a linear algebra class? If so, you should have covered this; if not, then I presume you haven't studied linear algebra, so I'm not sure why you'd want to find this inverse.
 
  • #6
Ok, I'll look.
 
Last edited:
  • #7
Thanks. That was very helpful.
 
  • #8
This is quiet an easy thing to do. Row operations are many but the most basic ones are addition of two rows, substitution and multiplication by a scalar.
Your goal is to make look like
1111 1000
1113 0100
1133 0010
1333 0001
I would suggest that you subtract the first row from every other row making all elements in the first column underneath the first entry zeros.
Do the same for each column to reduce your matrix to row echelon form.
Remember to execute the same operations on your identity matrix as well!

Good luck.
 

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

The inverse of a matrix is used to solve systems of linear equations, which is an essential tool in many scientific and mathematical applications. It allows us to find the unique solution to a system of equations, and also makes it possible to perform operations such as division on matrices.

2. How is the inverse of a matrix calculated?

The inverse of a matrix is calculated by using a specific formula, known as the Gauss-Jordan elimination method. This involves performing a series of elementary row operations on the matrix until it is in reduced row-echelon form. The resulting matrix is the inverse of the original matrix.

3. Is every matrix invertible?

No, not every matrix is invertible. A matrix must be square (same number of rows and columns) and have a non-zero determinant in order to be invertible. If the determinant is zero, the matrix is not invertible and is known as a singular matrix.

4. Can the inverse of a matrix be used to solve any system of equations?

The inverse of a matrix can only be used to solve systems of linear equations. Non-linear systems of equations cannot be solved using the inverse of a matrix.

5. Are there any properties of the inverse of a matrix?

Yes, there are a few important properties of the inverse of a matrix. For example, the inverse of a matrix multiplied by the original matrix will result in the identity matrix. Also, the inverse of the inverse of a matrix is the original matrix. Additionally, a matrix and its inverse commute, meaning the order of multiplication does not matter.

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