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  • If the matrices <img alt="A=\begin{bmatrix} 1 &1 &2 \\ 1 &3 &4 \\ 1& -1 &3 \end{bmatrix}" src="https://entrancecorner.oncodecogs.com/gif.latex?A%3D%5Cbegin%7Bbmatrix%7D

If the matrices A=\begin{bmatrix} 1 &1 &2 \\ 1 &3 &4 \\ 1& -1 &3 \end{bmatrix}, B=adj \: A and C=3A, then \frac{\left | adj\: B \right |}{\left | C \right |}  is equal to : 

Option: 1

16


Option: 2

2


Option: 3

8


Option: 4

72


Answers (1)

best_answer

 

 

Properties of adjoint of Matrix - Part 2 -

Properties of adjoint of matrix

5.  If A is non-singular square matrix, then, \mathrm{|adj (adj A)|= |A|^{(n-1)^2}}

      Proof: from the previous property, we know that 

       \\\mathrm{adj (adj A)= |A|^{(n-2)}A} \\\mathrm{taking\; determinant\; on \; both\; sides, then} \\\mathrm{|adj (adj A)|=||A|^{(n-2)}A|} \mathrm{=|A|^{n(n-2)}|A| = |A|^{(n-1)^2}}\\\left ( \text{using}\;\;|kA|=k^n|A| \right )

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|A|=\left|\begin{array}{ccc}{1} & {1} & {2} \\ {1} & {3} & {4} \\ {1} & {-1} & {3}\end{array}\right|=6

|\operatorname{adj} \mathrm{B}|=\left.|\operatorname{adj\;adj} \mathrm{A}|=| \mathrm{A}\right|^{(n-1)^{2}}=|\mathrm{A}|^{4}=(36)^{2}

{|C|=|B A|=3^{3} \times 6} \\ {\frac{|\operatorname{adj B}|}{|C|}=\frac{36 \times 36}{3^{3} \times 6}=8}

Posted by

Divya Prakash Singh

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