If [A = [ pmatrix 5a & - b 3 & 2 pmatrix ]] and A adj A=A A^T, then 5a+b is equal to :

Question

If $$A = \left[ {\begin{pmatrix} {5a} & { - b}  \
   3 & 2  \ \end{pmatrix} } \right]$$  and $A$ adj $A=A$ ${A^T},$ then $5a+b$ is equal to :

Accepted Answer

$$A\left( {Adj\,\,A} \right) = A\,{A^T}$$

$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$

$$Adj\,\,A = {A^T}$$

$$ \Rightarrow \left[ {\begin{pmatrix} 2 & b  \
   { - 3} & {5a}  \

\end{pmatrix} } \right] = \left[ {\begin{pmatrix} {5a} & 3  \
   { - b} & 2  \

\end{pmatrix} } \right]$$

$$ \Rightarrow a = {2 \over 5}\,\,$$ and $$\,\,b = 3$$

$$ \Rightarrow 5a + b = 5$$

Solutions · 1