Elementary Row Matrix Calculator

Instructions: Use this calculator to generate an elementary row matrix that will multiply row $$p$$ by a factor $$a$$, and row $$q$$ by a factor $$b$$, and will add them, storing the results in row $$q$$. Please provide the required information to generate the elementary row matrix.

The notation you follow is $$a R_p + b R_q \rightarrow R_q$$

Size of the matrix $$n$$ (Integer. Ex: 2, 3, 4, etc.)
Row that receives the result $$q$$ (Integer. Ex: 2, 3, 4, etc.)
Factor $$b$$ that multiplies row $$q$$ (Ex: 2, 3/4, -1, etc.)
The other row $$p$$ (Integer. Ex: 2, 3, 4, etc.)
Factor $$a$$ that multiplies row $$p$$ (Ex: 2, 3/4, -1, etc.)

Elementary row matrices are crucial matrices that have a very important property: when multiplying a matrix by them, the result is that the matrix essentially preserves all its rows, except for one, which stores the operation between two rows of the matrix.

Notationwise, there are several ways to name these type of matrices. One notation is $$E_{p,q}(a, b)$$, which indicates an elementary matrix that multiplies row $$p$$ by $$a$$, row $$q$$ by $$b$$, add these two and it stores the result on row $$q$$.

Another way of expressing the same is: $$b R_{q} + a R_{p} \rightarrow R_q$$. Now, why would we even define this matrix? Because it is SUPER useful, for reduced getting the reduced row echelon form, for example.

How do you calculate elementary row operations?

That is the magic of elementary row matrices: they are able to conduct matrix row operations by multiplying the given matrix by a certain elementary matrix. And one thing that is super neat is that elementary matrices are invertible.

Elementary row operations inverse calculator

One of the most important applications of elementary row matrices is for computing inverses. You start with a given matrix $$A$$, and you augment it with the identity matrix, so you have an augmented matrix $$[A | I]$$.

Using appropriate elementary row matrices, you obtain the row-echelon form. If you have a perfect echelon form (with all the subdiagonal different from zero, then the matrix is invertible.

You continue conducting row echelon reduction upwards, until you have converted the original matrix into the identity $$I$$. The resulting augmented part, that has captured all the elementary matrices, is the inverse $$A^{-1}$$.