Question

If P = \[\left[ {\begin{array}{*{20}{c}}
  1&\alpha &3 \\
  1&3&3 \\
  2&4&4
\end{array}} \right]\]is the adjoint of a 3 × 3 matrix A and |A| = 4, then α is equal to:
$
  {\text{A}}{\text{. 11}} \\
  {\text{B}}{\text{. 5}} \\
  {\text{C}}{\text{. 0}} \\
  {\text{D}}{\text{. 4}} \\
$

Answer 1

Hint: In order to find the value of α from the given we apply the concept of adjoint of a matrix. Adjoint of a matrix is nothing but the transpose of the co-factor matrix of the given matrix. We first find the determinant of the matrix P and apply the formula expressing a relation between the adjoint and determinant of a matrix.

Complete step-by-step answer:
Given Data,
P = \[\left[ {\begin{array}{*{20}{c}}
  1&\alpha &3 \\
  1&3&3 \\
  2&4&4
\end{array}} \right]\]
|A| = 4

We know the determinant of a matrix of the form X = \[\left[ {\begin{array}{*{20}{c}}
  {\text{a}}&{\text{b}}&{\text{c}} \\
  {\text{d}}&{\text{e}}&{\text{f}} \\
  {\text{g}}&{\text{h}}&{\text{i}}
\end{array}} \right]\]is given by the formula,
Det X = [a (e × i – f × h) – b (d × i – f × g) + c (d × h – e × g)]
Using this formula we can determine the determinant of the adjoint matrix of A, which is P as follows:
Det P also represented as |P|
|P| = 1 (12 – 12) – α (4 – 6) + 3 (4 – 6)
⟹|P| = 2α – 6 -- (1)

Given that the matrix P is the adjoint of matrix A, P = adj A

We know the formula relating the adjoint of a matrix to its determinant is given by,
For a matrix Y, we know
$|{\text{adj Y| = |Y}}{{\text{|}}^{{\text{n - 1}}}}$
The determinant of the adjoint matrix is equal to the determinant of the given matrix raised to the power of ‘n – 1’ where n is the value of the order of the matrix.
Given a 3 × 3 matrix, hence n = 3
Also given P is the adjoint of the matrix A, hence |P| = |adj A|

We substitute the values of |P| from equation (1) and the value of determinant of A given in the question in the above formula, we write
$|{\text{adj A| = |A}}{{\text{|}}^{{\text{3 - 1}}}}$
$ \Rightarrow 2\alpha {\text{ - 6 = }}{{\text{4}}^2}$
$ \Rightarrow 2\alpha {\text{ - 6 = 16}}$
$ \Rightarrow 2\alpha {\text{ = 22}}$
$ \Rightarrow \alpha {\text{ = 11}}$

Hence α is equal to 11.
Option A is the correct answer.

Note: In order to solve this type of questions the key is to know the concepts involved in matrices. We need to know the meaning and the formula to find out the concepts of a matrix such as the determinant, adjoint, cofactors, inverse of a matrix etc. Order of the matrix is defined as the number of rows by the number of columns a matrix has. It is not compulsory that the order of a matrix has to be uniform.