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`A^(-1) = f( A )`

Enter a value for all fields

The **Inverse of a 3x3 Matrix** calculator computes the matrix (A^{-1}) that is the inverse of the base matrix (A).

**INSTRUCTIONS:** Enter the following:

- (
**A**) 3x3 matrix - (
**n**) Number of decimals for rounding.

**Inverse Matrix (A ^{-1}):** The calculator returns the inverse matrix (A

- Determinant of 3-by-3 Matrix
- Characteristic Polynomial of a 3x3 matrix
- Product of a 3x3 matrix and a 3x1 matrix
- Inverse of a 3x3 Matrix
- Transpose of a 3x3 Matrix
- Trace of a 3x3 Matrix
- Mirror of a 3x3 Matrix
- Cramer's Rule (three equations, solved for x)
- Cramer's Rule (three equations, solved for y)
- Cramer's Rule (three equations, solved for z)
- Cramer's Rule Calculator

An invertible matrix is a matrix that is square and nonsingular. In Linear Algebra, the inverse of a given matrix relates well to Gaussian elimination; you may wish to visit what it means to perform elementary row operations by going to our page on the Row Echelon Form of a 3x3 matrix.

A square matrix is a matrix that has the same number of rows and columns, often referred to as an `n times n` matrix. In the case above, we are taking the inverse of a `3 times 3` matrix, where there are three rows and three columns. A nonsingular matrix is a matrix whose determinant is not equal to zero; a singular matrix is not invertible because it will not reduce to the identity matrix.

When it comes to the basic idea of an inverse, it is explained by Williams in the following manner (69):

Suppose you have two numbers such that

`a*b=1` and `b*a=1`

this means that a and b are multiplicative inverses of each other.

For example, take `a=frac(1)(5)` and `b=5.` It is clear that when you multiply `frac(1)(5) * 5` you get `1`.

The idea of a multiplicative inverse extends to matrices, where two matrices are inverses of each other if they multiply to get the identity matrix. The identity matrix for a `3 times 3` matrix is:

`I_(n)=[(1, 0 , 0),(0, 1, 0), (0, 0, 1)]`

On page 69, Williams defines the properties of a matrix inverse by stating, "Let `A` be an `n times n` matrix. If a matrix `B` can be found such that `AB = BA = I_(n)`, then `A` is said to be invertible and `B` is called an inverse of `A`. If such a matrix does not exist, then `A `has no inverse."

The inverse of a matrix relates to Gaussian elimination in that if you keep track of the row operations that you perform when reducing a matrix into the identity matrix and simultaneously perform the same operations on the identity matrix you end up with the inverse of the matrix you have reduced. This is particularly important to note because it extends to matrices of all different sizes since the identity matrix for an arbitrary `n times n` matrix always exists.

However, there is also a formulaic way of producing the inverse of a `3 times 3` matrix, which we will present below.

`A^(-1) = frac(1) (abs(A))[ (abs((A_(22), A_(23)), (A_(32), A_(33))), abs((A_(13), A_(12)), (A_(33), A_(32))), abs((A_(12), A_(13)), (A_(22), A_(23)))), (abs((A_(23), A_(21)), (A_(33), A_(31))), abs((A_(11), A_(13)), (A_(31), A_(33))), abs((A_(13), A_(11)), (A_(23), A_(21)))), (abs((A_(21), A_(22)), (A_(31), A_(32))), abs((A_(12), A_(11)), (A_(32), A_(31))), abs((A_(11), A_(12)), (A_(21), A_(22))))]`,

where `abs(A)` is `det A`.

Williams, Gareth. Linear Algebra With Applications. Boston: Jones and Bartlett, 2011. Print.