If [P = [ pmatrix 1 & & 3 1 & 3 & 3 2 & 4 & 4 pmatrix ]] is the adjoint of a [3 3] matrix …

Question

If $$P = \left[ {\begin{pmatrix} 1 & \alpha  & 3  \
   1 & 3 & 3  \
   2 & 4 & 4  \ \end{pmatrix} } \right]$$ is the adjoint of a $$3 \times 3$$ matrix $A$ and

$$\left| A \right| = 4,$$ then $$\alpha $$ is equal to :

Accepted Answer

$$\left| P \right| = 1\left( {12 - 12} \right) - \alpha \left( {4 - 6} \right) + $$

$$\,\,\,\,\,\,\,\,\,\,\,3\left( {4 - 6} \right) = 2\alpha  - 6$$

Now, $$adj\,\,A = P\,$$ $$\,\,\,\,\,\,\,\, \Rightarrow \left| {adj\,A} \right| = \left| P \right|$$

$$ \Rightarrow {\left| A \right|^2} = \left| P \right|$$

$$ \Rightarrow \left| P \right| = 16$$

$$ \Rightarrow 2\alpha  - 6 = 16$$

$$ \Rightarrow \alpha  = 11$$

Solutions · 1