Question

If the adjoint of a 3×3 matrix P is $\,\left[ {\begin{array}{*{20}{c}}
  1&4&4 \\
  2&1&7 \\
  1&1&3
\end{array}} \right]$ then the possible value(s) of the determinant of P is(are)?

A. -2
B. -1
C. 1
D. 2

Answer 1

Hint: In order to solve this problem we need to know that if we have the determinant of a matrix then we can get the determinant of its adjoint and vice versa using the formula ${\left| A \right|^{n - 1}} = \left| {adjA} \right|$ where n is the order of the matrix. Knowing this will solve your problem.

Complete step-by-step answer:
We have adjoint of a matrix P that is adjP = $\,\left[ {\begin{array}{*{20}{c}}
  1&4&4 \\
  2&1&7 \\
  1&1&3
\end{array}} \right]$
We know that if there is a matrix A then ${\left| A \right|^{n - 1}} = \left| {adjA} \right|$ n is the order of the matrix. Doing the same thing for above matrix we get,
We can say that $\left| {adjP} \right| = {\left| P \right|^{n - 1}}$…………..(1)
The value of $\left| {adjP} \right|$ can be found as,
\[
   \Rightarrow \,\left[ {\begin{array}{*{20}{c}}
  1&4&4 \\
  2&1&7 \\
  1&1&3
\end{array}} \right] \\
   \Rightarrow 1\left( {3 - 7} \right) - 4\left( {6 - 7} \right) + 4\left( {2 - 1} \right) \\
   \Rightarrow - 4 + 4 + 4 = 4 \\
    \\
\]
So, we get the determinant of adjP = 4
Now, using 1 we can say that $4 = {\left| P \right|^{3 - 1}} = {\left| P \right|^2}$
So, the value of the determinant of P can be calculated as,
$
   \Rightarrow {\left| P \right|^2} = 4 \\
   \Rightarrow \left| P \right| = \pm \sqrt 4 = \pm 2 \\
   \Rightarrow \left| P \right| = \pm 2 \\
$
P is in mod therefore the value of P can be +2 or -2.
So, the correct options are D and A.

Note: When you get to solve such problems you need to know the only formula ${\left| A \right|^{n - 1}} = \left| {adjA} \right|$ where n is the order of the matrix. We also need to know that the adjoint of the matrix is the transpose of the cofactor of the matrix. You can also find the inverse of the matrix with the help of adjoint of the matrix. For finding the inverse of a 3×3 matrix, first of all, calculate the determinant of the matrix and id the determinant is 0 then it has no matrix. After that, rearrange the matrix by rewriting the first row as the first column, middle row as middle column and final row as the final column. Knowing this will help you further and will give you the right answers.