Invertible Matrix Calculator

Instructions: Use this invertible matrix calculator to determine whether a given matrix is invertible or not, showing all the steps. First, click on one of the buttons below to specify the dimension of the matrix you want to assess invertibility.

Then, click on the first cell and type the value, and move around the matrix by pressing “TAB” or by clicking on the corresponding cells, to define ALL the matrix values.

## Invertible Matrix Calculator

One of the center elements in Linear Algebra is the concept of a matrix. Matrices are arrays of numbers organized in rows and columns.

Matrix operations can be intuitively defined, especially when you sum or subtract matrices, which in the end all your doing is adding and subtracting component by component.

The idea of multiplication of matrices is slightly less intuitive for the non initiated but, you gotta trust me here, there are good reasons why the matrix multiplication is defined in the way it is.

### What do you use the inverse matrix for?

- When a matrix is invertible, you can compute its inverse
- You can use the inverse to move freely the matrix “to the other side of the equation”
- This allows you to simply solve a system of equations by finding inverse of a matrix

### What is the inverse of a Matrix?

Square matrices (this is, matrices that have the same number of rows and columns) can be invertible or not.

For a matrix \(A\) to be invertible means that there is another matrix \(B\) such that the product of \(A\) and \(B\) equals the identity matrix ( a special matrix with one’s in the diagonal, and zeroes outside of the diagonal).

Why would you be interested in whether a matrix is invertible or not, you may ask? Good question. When the matrix is invertible, we can “pass the matrix to the other side”, the same way you would do in a simple equation with numbers.

In this case, you can find the inverse of the matrix and you “pass” the inverse of the matrix to the other side of the equation

In practical terms, if you have an matrix equation \( Ax = b \), and \(A\) is invertible, then the equation has a unique solution, which can be written as \(x = A^{-1} b\), where \(A^{-1}\) is the inverse matrix of A, under the assumption it exists.

### When a matrix is invertible?

There are many, many ways to characterize whether or not a matrix is invertible. You can apply different “tests” to tell whether a matrix is invertible or not. The test you choose will sometimes depend on the structure of the matrix.

One commonly used test to assess whether a matrix is invertible is to first compute the determinant of the matrix. If the determinant is different than zero, then the matrix is invertible. But then if it is zero, then the matrix is NOT invertible. Pretty simple, huh?

### Is a matrix invertible 3×3? How to know

First, since 3×3 is a square matrix, it is a candidate to check for its invertiblity (non-square matrices are discarded right away)

### Are all 2×2 matrices invertible?

Not at all. There are lots of 2×2 matrices that are not invertible. For example, the matrix

\[ A = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix}\]

is a simple example of a 2×2 matrix that is not invertible.

How do you know if a matrix is invertible without determinant?

As we said earlier, there are many tests to assess whether or not a matrix is invertible, and not all methods use the determinant

One method to do is to use Gauss-method (using the operation of elementary matrices) to convert the matrix into row-echelon form, and once that is done, you take a look at the diagonal of the row-echelon form: if all the diagonals are non-zero, then the matrix is invertible, and if ANY element in the diagonal of the echelon-form is zero, then the matrix is not invertible.

### Example: Invertibiliy of a Matrix

Question: Assume that you have the following matrix:

\[ \begin{bmatrix}2&1&2\\1&4&1\\2&1&3\end{bmatrix}\]

Solution: We need to determine whether the \(3 \times 3\) matrix that has been provided is invertible or not.

### Step 1: Method Used

There are several methods to determine whether a matrix is invertible or not. The method we will use in this case is the method of the determinant.

Put very simply, we will compute the determinant, and if the determinant is different from zero, then the matrix is invertible, but it is equal to zero, then the matrix is not invertible.

### Step 2: Calculation of the Determinant

Using the sub-determinant formula we get:

\[ \begin{vmatrix} \displaystyle 2&\displaystyle 1&\displaystyle 2\\[0.6em]\displaystyle 1&\displaystyle 4&\displaystyle 1\\[0.6em]\displaystyle 2&\displaystyle 1&\displaystyle 3 \end{vmatrix} = 2 \cdot \left( 4 \cdot \left( 3 \right) – 1 \cdot \left(1 \right) \right) – 1 \cdot \left( 1 \cdot \left( 3 \right) – 2 \cdot \left(1 \right) \right) + 2 \cdot \left( 1 \cdot \left( 1 \right) – 2 \cdot \left(4 \right) \right)\] \[ = 2 \cdot \left( 11 \right) – 1 \cdot \left( 1 \right) + 2 \cdot \left( -7 \right) = 7\]

### Step 3: Conclusion

We conclude that since \(\det A = \displaystyle 7 \ne 0\), then the given matrix is invertible.