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Question: If A is non-singular matrix of order 3 and \({\text{|adjA| = |A}}{{\text{|}}^{\text{K}}}\), then wri...

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

Explanation

Solution

Hint: In order to solve this problem one must know the formula |adjA| = |A|n - 1{\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 |adjA| = |A|K{\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 |adjA| = |A|n - 1{\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 |adjA| = |A|K = |A|n - 1=|A|3−1 = |A|2{\text{|adjA| = |A}}{{\text{|}}^{\text{K}}}{\text{ = |A}}{{\text{|}}^{{\text{n - 1}}}} = {\text{|A}}{{\text{|}}^{3 - 1}}{\text{ = |A}}{{\text{|}}^2}
Then we get |A|K=|A|2{\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 |adjA| = |A|n - 1{\text{|adjA| = |A}}{{\text{|}}^{{\text{n - 1}}}} where n is the order of the matrix.