Inverse $T$ matrix of a 3*3 matrix.

Question

I have the matrix $$ A= \begin{bmatrix} 1 & 4 & 1\\ 0 & 2 & 5\\ 0 & 0 & 5 \end{bmatrix}. $$ I have found the $$ T= \begin{bmatrix} 1 & 4 & 1\\ 0 & 0 & 1\\ 0 & 1 & 3/5 \end{bmatrix}. $$ How can I find the $T^{-1}$? I already know the $\frac{1}{|T|}$ part but I am confused with the adjugate matrix.

Answer 1

I think the easiest way (other than asking Wolfram Alpha) is to do a row echelon reduction on the augmented matrix

$$ U = \left[ \begin{array}{ccc|ccc} 1 & 4 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 3/5 & 0 & 0 & 1 \\ \end{array} \right] $$

This gives

$$ V = \left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 7/5 & \mbox{-}4 \\ 0 & 1 & 0 & 0 & \mbox{-}3/5 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ \end{array} \right] $$

which means the inverse is

$$ T^{-1} = \left[ \begin{array}{ccc} 1 & 7/5 & \mbox{-}4 \\ 0 & \mbox{-}3/5 & 1 \\ 0 & 1 & 0 \\ \end{array} \right]$$