Elementary Matrices and Elementary Row Operations

  
  a={{1,2,-4,6},{2,-4,6,1},{5,6,12,-2},{1,1,4,-3}};
  b={{1,4,-4},{1,2,8},{0,3,2},{1,-4,0}};
  MatrixForm[a]
  MatrixForm[b]

  a.IdentityMatrix[4];
  MatrixForm[%]

  b.IdentityMatrix[3];
  MatrixForm[%]

What happens when you left multiply a matrix by the identity matrix?

  IdentityMatrix[4].a;
  MatrixForm[%]

  IdentityMatrix[4].b;
  MatrixForm[%]

I hope this was not too difficult, and that you can generalize this idea.

Let's exchange rows of the identity matrix and multiply by the new matrix. What happens?

  MatrixForm[E1[1,2,3]]
  MatrixForm[E1[1,3,3]]
  MatrixForm[E1[2,3,3]]

Let's left multiply first:

  Clear[a,b]
  a={{1,2,-1},{2,-5,2},{8,4,-3}};
  b={{3,2,-2},{10,-3,4},{6,-10,4},{4,6,1}};
  c={{1,6},{6,2},{5,-2}};
  MatrixForm[a]
  MatrixForm[b]
  MatrixForm[c]

  MatrixForm[a]
  E1[1,2,3].a;
  MatrixForm[%]

  MatrixForm[c]
  E1[1,3,3].c;
  MatrixForm[%]

Let's right multiply:

  MatrixForm[a]
  a.E1[2,3,3];
  MatrixForm[%]

  MatrixForm[b]
  b.E1[1,2,3];
  MatrixForm[%]

I hope you can generalize these ideas to nxn matrices. The matrices E1 are called permutation matrices or sometimes type I elementary matrices.

Let's multiply the rows of the identity matrix by some number and multiply by the new matrix. What happens?

  MatrixForm[E2[2,4,3]]

  MatrixForm[E2[3,-2,3]]

Let's left multiply first:

  MatrixForm[a]
  E2[1,3,3].a;
  MatrixForm[%]

  MatrixForm[c]
  E2[2,-1/2,3].c;
  MatrixForm[%]

Let's right multiply:

  MatrixForm[a]
  a.E2[3,1/4,3];
  MatrixForm[%]

  MatrixForm[b]
  b.E2[2,2,3];
  MatrixForm[%]

I hope you can generalize these ideas to nxn matrices. The matrices E2 are called type II elementary matrices.

Now we replace a row in the identity matrix by its sum with a multiple of another row.

  MatrixForm[E3[1,2,2,3]]
  MatrixForm[E3[3,2,-2,3]]

Let's left multiply first:

  MatrixForm[a]
  E3[1,2,2,3].a;
  MatrixForm[%]

  MatrixForm[c]
  E3[3,2,-2,3].c;
  MatrixForm[%]

Let's right multiply:

  MatrixForm[a]
  a.E3[1,3,2,3];
  MatrixForm[%]

  MatrixForm[b]
  b.E3[2,1,10,3];
  MatrixForm[%]

This one is a bit harder to see, but I hope you can generalize these ideas to nxn matrices. The matrices E3 are called type III elementary matrices.

The three types of operations described above and represented by the three types of elementary matrices are called Elementary Row Operations. If it is possible to get from matrix A to matrix B (and as we'll see next, vice versa) using only a finite number of these elementary row operations, then we say A and B are Row Equivalent.

Up to Linear Algebra Part I