Elimination of Matrices

We have an example $A\mathbf{x}=\mathbf{b}$,

$A=\left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]$ and $\mathbf{b}=\left[\begin{array}{c}2\\ 12\\ 2\end{array}\right]$.

Steps of Elimination:

• Step 1: subtract $3$ times row 1 from row 2;
• Step 2: subtract $2$ times row 2 from row 3.

$$A=\left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]\to \left[\begin{array}{ccc}1& 2& 1\\ 0& 2& -2\\ 0& 4& 1\end{array}\right]\to U=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& -2\\ 0& 0& 5\end{array}\right]$$

$$\mathbf{b}=\left[\begin{array}{c}2\\ 12\\ 2\end{array}\right]\to \cdots \to \left[\begin{array}{c}2\\ 6\\ -10\end{array}\right]$$

From row3, $5z=-10$, so $z=-2$

Thus, we can easily solve the systems of equations, $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ -2\end{array}\right]$.

Elimination Matrices §

The product of a matrix (3x3) and a column vector (3x1) is a column vector (3x1) that is a linear combination of the columns of the matrix.

The product of a row vector (1x3) and a matrix (3x3) is a row vector (1x3) that is a linear combination of the rows of the matrix.

For example,

$$\left[\begin{array}{ccc}1& 0& 0\\ -3& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 1\\ 3& 8& 1\\ 0& 4& 1\end{array}\right]=\left[\begin{array}{ccc}1& 2& 1\\ 0& 2& -2\\ 0& 4& 1\end{array}\right].$$

Multiplying on the left by a permutation matrix exchanges the rows of a matrix, while multiplying on the right exchanges the columns. For example,

$$P=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right].$$

$P$ is a Permutation Matrix and the first and second rows of the matrix $PA$ are the second and first rows of the matrix $A$.

Note, matrix multiplication is associative but not commutative.