Question

If the value of a determinant of third order is 12, then the value of the determinant formed by replacing each element by its co-factor will be …​

Answer 1

From property of cofactor, we have

Let A be any matrix of order n, then

The matrix obtained say P, by replacing each element of matrix A by its Co factor has a property that

$\sf \: | P | \: = \: |A | {}^{n - 1}$

So, by applying this property,

$\sf \: | P | \: = \: |12 | {}^{3 - 1}$

$\sf \: | P | \: = \: |12 | {}^{2}$

$\sf \: | P | \: = \: 144$

Therefore,

If the value of a determinant of third order is 12, then the value of the determinant formed by replacing each element by its co-factor will be 144.

Answer 2

Given,

Determinant of matrix,$\[|A| = 12\]$

Order of matrix, n $\[ = 3\]$

Solution,

Calculate the determinant formed by replacing each element by its co-factor.

$\[cof|A| = |A{|^{n - 1}}\]$

$\[\begin{array}{l} \Rightarrow cof|A| = {\left( {12} \right)^{3 - 1}}\\ \Rightarrow cof|A| = {\left( {12} \right)^2}\\ \Rightarrow cof|A| = 144\end{array}\]$

Hence the determinant of matrix formed by replacing each element by its co-factor is $\[144\]$ .