written 7.3 years ago by | • modified 4.0 years ago |
$\text {Is reduced to normal form Also find rank }.$
written 7.3 years ago by | • modified 4.0 years ago |
$\text {Is reduced to normal form Also find rank }.$
written 7.3 years ago by |
In the form of A=IAI
i.e.
$\begin{vmatrix} 1&2&3&4\\ 2&1&4&3\\ 3&0&5&-10\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\0&1&0\\0&0&1 \\ \end{vmatrix}A\begin{vmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \\ \end{vmatrix}$
By $R_2-2R_1$ and $R_3-3R_1$
$\begin{vmatrix} 1&2&3&4\\ 0&-3&-2&-5\\ 0&-6&-4&-22\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\-3&0&1 \\ \end{vmatrix}A\begin{vmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \\ \end{vmatrix}$
By $C_2-2C_1, C_3-3C_1, C_4-4C_1$
$\begin{vmatrix} 1&0&0&0\\ 0&-3&-2&-5\\ 0&-6&-4&-22\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\-3&0&1 \\ \end{vmatrix}A\begin{vmatrix} 1&-2&-3&-4\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \\ \end{vmatrix}$
By $R_3-2R_2$
$\begin{vmatrix} 1&0&0&0\\ 0&-3&-2&-5\\ 0&-3&-2&-5\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\1&-2&1 \\ \end{vmatrix}A\begin{vmatrix} 1&-2&-3&-4\\0&1&0&0\\0&0&1&0\\ 0&0&0&1 \\ \end{vmatrix}$
By $ \dfrac {-1}3C_2, \dfrac {-1}2C_3, \dfrac {-1}5C_5$
$\begin{vmatrix} 1&0&0&0\\ 0&1&1&1\\ 0&0&0&12/5\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\1&-2&1 \\ \end{vmatrix}A\begin{vmatrix} 1&2/3&3/2&4/5\\0&-1/3&0&0 \\ 0&0&1/2&0 \\ 0&0&0&-1/5 \\ \end{vmatrix}$
By $C_3-C_2 ,C_4-C_3$
$\begin{vmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&0&12/5\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\1&-2&1 \\ \end{vmatrix}A\begin{vmatrix} 1&3/3&3/6&-7/10 \\ 0&-1/3&1/3&0 \\ 0&0&1/2&-1/2 \\ 0&0&0&-1/5 \\ \end{vmatrix}$
By $C_4\times \dfrac 5{12} $ and $C_{34}$
$\begin{vmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ \end{vmatrix}=\begin{vmatrix} 1&0&0\\-2&1&0\\1&-2&1 \\ \end{vmatrix}A\begin{vmatrix} 1&2/3&-7/24&5/6 \\ 0&-1/3&0&1/3 \\ 0&0&-5/24&1/2 \\ 0&0&-1/12&0 \\ \end{vmatrix}$
Rank of A=3
$P= \begin{vmatrix} 1&0&0\\-2&1&0\\1&-2&1 \\ \end{vmatrix} $ & $Q=\begin{vmatrix} 1&2/3&-7/24&5/6 \\ 0&-1/3&0&1/3 \\ 0&0&-5/24&1/2 \\ 0&0&-1/12&0 \\ \end{vmatrix}$