Question

Solve the following :

If A = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 3& 1\end{array}\right]}$, the reduce it to unit matrix by using row transformations.

Answer 1

AB = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 3& 1\end{array}\right]}$

∴ |A| = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 3& 1\end{array}\right]}$

= 1(1 – 0) –0(2 –0) + 0(6 – 3)
= 1 – 0 + 0
= 1 ≠ 0
∴ A is non-singular matrix.
Hence, row transformations are possible.

Now, A = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 3& 1\end{array}\right]}$

Applying R₂ → R₂ – 2R₁ and R₃ → R₃ – 3R₁, we get

A = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 3& 1\end{array}\right]}$

Applying R₃ → R₃ – 3R₂, we get

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