High School Mathematics Extensions/Matrices/Project/Elementary Matrices

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

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

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

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

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

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

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

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 C^T where the ith row of C is the ith column of C^T. Prove using elementary matrices that

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

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?