Question

If A is non-singular matrix of order 3 and ${\text{|adjA| = |A}}{{\text{|}}^{\text{K}}}$, then write the value of K.

Answer 1

Hint: In order to solve this problem one must know the formula ${\text{|adjA| = |A}}{{\text{|}}^{{\text{n - 1}}}}$ where n is the order of the matrix. Using this will solve our problem and we will get the right value of K.

Complete step-by-step answer:
The given equation is ${\text{|adjA| = |A}}{{\text{|}}^{\text{K}}}$ A is a matrix of order 3.
And we know that if A is a matrix of order 3 then ${\text{|adjA| = |A}}{{\text{|}}^{{\text{n - 1}}}}$ where n is the order of the matrix.
Here order is 3 so n = 3.
Then we can say that ${\text{|adjA| = |A}}{{\text{|}}^{\text{K}}}{\text{ = |A}}{{\text{|}}^{{\text{n - 1}}}} = {\text{|A}}{{\text{|}}^{3 - 1}}{\text{ = |A}}{{\text{|}}^2}$
Then we get ${\text{|A}}{{\text{|}}^{\text{K}}} = {\text{|A}}{{\text{|}}^2}$
Hence, the value of K is 2.

Note: Whenever you face such types of problems then you need to know the most important formula of matrices and determinants like ${\text{|adjA| = |A}}{{\text{|}}^{{\text{n - 1}}}}$ where n is the order of the matrix.