High School Mathematics Extensions/Matrices/Project/Elementary Matrices

Project -- Elementary matrices

Throughout, $A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}$

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

• ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$
• ${\begin{pmatrix}1&f\\0&1\end{pmatrix}}$  where f is a scalar
• ${\begin{pmatrix}1&0\\f&1\end{pmatrix}}$  where f is a scalar
• ${\begin{pmatrix}f&0\\0&1\end{pmatrix}}$  where f is a scalar
• ${\begin{pmatrix}1&0\\0&f\end{pmatrix}}$  where f is a scalar

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

• $B={\begin{pmatrix}0&1\\1&0\end{pmatrix}}A$
• $B={\begin{pmatrix}1&f\\0&1\end{pmatrix}}A$  where f is a scalar
• $B={\begin{pmatrix}1&0\\f&1\end{pmatrix}}A$  where f is a scalar
• $B={\begin{pmatrix}f&0\\0&1\end{pmatrix}}A$  where f is a scalar
• $B={\begin{pmatrix}1&0\\0&f\end{pmatrix}}A$  where f is a scalar

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

${\begin{pmatrix}1&2\\4&3\end{pmatrix}}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1&3\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\1&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&5\end{pmatrix}}{\begin{pmatrix}1&-3\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}$

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 CT where the ith row of C is the ith column of CT. 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?