Question

Find the adjoint and the inverse of the matrix
A=1 3 3
1 4 3
1 3 4​

Answer 1

The adjoint and inverse of the matrix

$A=\left[\begin{array}{ccc}1&3&3\\1&4&3\\1&3&4\end{array}\right]$   is $\left[\begin{array}{ccc}7&-3&-3\\-1&1&0\\-1&0&1\end{array}\right]$

Step-by-step explanation:

Given:

 $A=\left[\begin{array}{ccc}1&3&3\\1&4&3\\1&3&4\end{array}\right]$      

To find:

Adjoint and the inverse of the matrix A.

Solution:

$A=\left[\begin{array}{ccc}1&3&3\\1&4&3\\1&3&4\end{array}\right]$

$A^{-1} =\frac{1}{|A|} adj(A)$     { $|A|$≠$0$ }

$|A|= (16-9)-3(4-3)+3(3-4)$

$|A|=1$     { $1$≠$0$ }

To find an adjoint of $A$,

Co-factor of $1=16-9=7$

Co-factor of $3$ $=4-3=1=-1$   ( using the adjoint property )

Co-factor of $3$ $=3-4=-1$

Co-factor of $1$  $=\left[\begin{array}{ccc}3&3\\3&4\\\end{array}\right] =12-9=3=-3$    ( using the adjoint property )

Co-factor of $4$ $=\left[\begin{array}{ccc}1&3\\1&4\end{array}\right] =4-3=1$

Co-factor of $3$ $=\left[\begin{array}{ccc}1&3\\1&3\end{array}\right] =3-3=0$

Co-factor of $1$  $=\left[\begin{array}{ccc}3&3\\4&3\end{array}\right] =9-12=-3$

Co-factor of $3$ $=\left[\begin{array}{ccc}1&3\\1&3\end{array}\right] =3-3=0$

Co-factor of $4$ $=\left[\begin{array}{ccc}1&3\\1&4\end{array}\right] =4-3=1$

$adj(A)=\left[\begin{array}{ccc}7&-1&-1\\-3&1&0\\-3&0&1\end{array}\right]$

Now transport the matrix,

⇒ $adj(A)=\left[\begin{array}{ccc}7&-3&-3\\-1&1&0\\-1&0&1\end{array}\right]$  

$A^{-1} =\frac{1}{|A|} adj(A)$

⇒  $A^{-1}= \left[\begin{array}{ccc}7&-3&-3\\-1&1&0\\-1&0&1\end{array}\right]$

So the adjoint and the inverse of the matrix A is  

$\left[\begin{array}{ccc}7&-3&-3\\-1&1&0\\-1&0&1\end{array}\right]$  .