High School Mathematics Extensions/Matrices/Project/Elementary Matrices

Project -- Elementary matrices

Throughout, $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

1. The matrices below are called elementary matrices. How are the matrices below different from the identity matrix I, describe each one.

• $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$
• $\left(\begin{array}{cc}1& f\\ 0& 1\end{array}\right)$ where f is a scalar
• $\left(\begin{array}{cc}1& 0\\ f& 1\end{array}\right)$ where f is a scalar
• $\left(\begin{array}{cc}f& 0\\ 0& 1\end{array}\right)$ where f is a scalar
• $\left(\begin{array}{cc}1& 0\\ 0& f\end{array}\right)$ where f is a scalar

2. In each of the cases, compute B then describe how is B different from A

• $B=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)A$
• $B=\left(\begin{array}{cc}1& f\\ 0& 1\end{array}\right)A$ where f is a scalar
• $B=\left(\begin{array}{cc}1& 0\\ f& 1\end{array}\right)A$ where f is a scalar
• $B=\left(\begin{array}{cc}f& 0\\ 0& 1\end{array}\right)A$ where f is a scalar
• $B=\left(\begin{array}{cc}1& 0\\ 0& f\end{array}\right)A$ where f is a scalar

3. The matrix $\left(\begin{array}{cc}1& 2\\ 4& 3\end{array}\right)$ has determinant not equal to zero. We can decompose the matrix into products of elementary matrices pre-multiplying the identity:

$\left(\begin{array}{cc}1& 2\\ 4& 3\end{array}\right)=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}1& 3\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 5\end{array}\right)\left(\begin{array}{cc}1& -3\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

Now suppose det(A) ≠ 0, can A be expressed as the product of elementary matrices and the identity?

4. a) Show that every elementary matrix has an inverse. Hint: use determinant.

b) Prove that every invertible matrix (a matrix that has an inverse) is the product of some elementary matrices pre-multiplying the identity.

5. A transpose of a matrix C is the matrix C^T where the ith row of C is the ith column of C^T. Prove using elementary matrices that

$(DE{)}^T=E^T{D}^T$

for arbitrary matrices D and E.

6. Show that every invertible matrix is also the product of some elementary matrices post-multiplying the identity.

7. How about non-invertible matrices? What can you say about them?