Question

If | (adj A) | = 81, for 3 $\times$ 3 matrix, then det A is equal to

Answer 1

Before solving this question one should have prior knowledge about the matrix and determinants and remember to use the relation between the adjoint and determinant of a matrix i.e. $|adjA| = |A{|^{n - 1}}$, use this information to approach the solution.

Complete step-by-step answer:
According to the given information we have a matrix with unknown elements with order 3 whose ad joint is equal to 81
We know that there is a relation between adjoint and the determinant of a matrix which is given by $|adjA| = |A{|^{n - 1}}$ here n is the order of the given matrix
Substituting the values in the above equation we get
$81 = |A{|^{3 - 1}}$
81 = $|A{|^2}$
$\Rightarrow$ $|A|$ = $\sqrt {81}$
$\Rightarrow$ $|A|$ = 9
Therefore value of $|A|$ is equal to 9

Note: In the above question we used the concept of matrix which can be explained as the method or way of arranging the inputs in the rows and columns which are rectangular arranged. There are many types of matrix such as row matrix it includes elements that are arranged in a row there is no column exists, column matrix it have elements in column not in row, square matrix in this type of matrix element are equally arranged in column and rows, diagonal matrix in this type of square matrix where all the elements except the diagonal are zero, unit matrix is the matrix where only all diagonal elements are one all except diagonal all elements are 0 and so many.