# Finding Inverse Matrix by Definition

• shelovesmath
In summary, the process for finding the inverse of a matrix by definition involves multiplying the original matrix by a variable matrix and setting it equal to the identity matrix. This can then be solved by using substitution and basic algebra. It is important to refer to the entries of the matrices by their row and column numbers rather than using letters or numbers for labeling.

#### shelovesmath

Hi fellow mathies,

So, I'm wondering if the way to find the inverse of a matrix by definition (instead of using a special algorithm/tacking on the identity matrix and reducing, etc), is to multiply the matrix by a variable matrix and have it equal the identity matrix.

So, for example:

this matrix

3 0 0
0 -1 3
0 -3 -1

multiplied by this matrix

a b c
d e f
g h i

would equal

1 0 0
0 1 0
0 0 1

would be

entry a is 3a+0+0=1
entry b is 3b+0+0=0
entry c is 3c + 0 + 0 =0
entry d is 0 -d + 3g = 0
entry 3 is 0 -e + +3h = 1
etc.

and then solve by substitution and basic algebra.

Last edited:
Yes, that works. (Is that all you wanted to know?). I just wouldn't call the entries "a,b,c,d,3,...". I actually don't understand your labeling system (unless you meant e when you wrote 3). You should refer to the entries by row number and column number. For example, what you called "3" is row 2, column 2, i.e. the "22" entry of the product.

## 1. What is the definition of an inverse matrix?

An inverse matrix is a square matrix that, when multiplied by the original matrix, results in the identity matrix. In other words, the inverse matrix "undoes" the original matrix.

## 2. How do you find the inverse matrix of a given matrix?

To find the inverse matrix, you must first check that the given matrix is invertible, meaning it has a non-zero determinant. Then, you can use the following formula: inverse matrix = (1/determinant) * (adjugate matrix), where the adjugate matrix is the transpose of the cofactor matrix.

## 3. Is every matrix invertible?

No, not every matrix is invertible. A matrix is only invertible if its determinant is non-zero. If the determinant is zero, then the matrix has no inverse.

## 4. What is the purpose of finding the inverse matrix?

The inverse matrix is used to solve systems of linear equations and to perform other mathematical operations, such as finding the determinant and eigenvalues of a matrix. It is also important in computer graphics and cryptography.

## 5. Can the inverse matrix of a matrix be found using a different method?

Yes, there are other methods for finding the inverse matrix, such as using row operations or using software programs. However, the definition method is the most straightforward and commonly used method for finding the inverse matrix.