# Homework Help: Given a Matrix A, find a Product of Elementary Matrices that equals A

1. Feb 4, 2010

### WTFsandwich

1. The problem statement, all variables and given/known data
Given

$$A$$ = $$\left( \begin{array}{cc} 2 & 1 \\ 6 & 4 \end{array} \right)$$

a) Express A as a product of elementary matrices.
b) Express the inverse of A as a product of elementary matrices.

2. Relevant equations

3. The attempt at a solution

Using the following EROs

Row2 --> Row2 - 3 * Row1
$$E_{1}$$ = $$\left( \begin{array}{cc} 1 & 0 \\ -3 & 1 \end{array} \right)$$

Row1 --> 1/2 * Row1
$$E_{2}$$ = $$\left( \begin{array}{cc} 1/2 & 0 \\ 0 & 1 \end{array} \right)$$

Row1 --> Row1 - 1/2 * Row2
$$E_{3}$$ = $$\left( \begin{array}{cc} 1 & 0 \\ -2 & 1 \end{array} \right)$$

Multiplying all the Elementary matrices together, I got the Product

$$P$$ = $$\left( \begin{array}{cc} 2 & -1/2 \\ -3 & 1 \end{array} \right)$$

Which is A-1.

Last edited: Feb 4, 2010
2. Feb 4, 2010

### VeeEight

Note that the inverse of A-1 is A and also that given invertible A and B, (AB)-1=B-1A-1

You have E1E2....En=A-1 where Ei is an elementary matrix. So take the inverse of the whole thing

3. Feb 4, 2010

### WTFsandwich

If I'm understanding you correctly, I should take the inverses of all the elementary matrices and multiply those, and it should give me A?

Essentially, (E1E2....En)-1 = A

Last edited: Feb 4, 2010
4. Feb 4, 2010

### krtica

Hey Sandwich,

Think of the matrix A as being equivalent to an identity matrix of the same size, but just manipulated by elementary row operations.

Vee is right, because if you multiply the inverse of A by A's corresponding elementary matrices, the product is the identity matrix. Try it out.

A=I(E1E2....En)^(-1)

Hope that helps!