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If A is a 3-rowed square matrix such that |A| = 2, then then |adj(adj(adjA))| is?
- a)512
- b)64
- c)256
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If A is a 3-rowed square matrix such that |A| =2, then then|adj(adj(ad...
This question is part of UPSC exam. View all Electronics and Communication Engineering (ECE) courses
Most Upvoted Answer
If A is a 3-rowed square matrix such that |A| =2, then then|adj(adj(ad...
Solution:
Given, |A| = 2
We know that adj(adj A) = |A|^(n-2) A, where n is the order of matrix A.
Here, n = 3 (since A is a 3-rowed square matrix)
So, adj(adj A) = 2^(3-2) A = 2A
Now, let's calculate A^2.
A^2 = A.A
We know that |A.A| = |A|.|A| = |A|^2
So, |A.A| = 2^2 = 4
Now, let's calculate adj(A^2).
adj(A^2) = (A^2)^(-1) |A^2|
We know that |A^2| = |A|^3 = 2^3 = 8
Also, (A^2)^(-1) = (A^(-1))^2 (since A is invertible)
So, adj(A^2) = (A^(-1))^2 |A^2| = 8(A^(-1))^2
Now, let's calculate adj(adj A^2).
adj(adj A^2) = (adj A^2)^(-1) |adj A^2|
We know that |adj A^2| = |A^2|^(n-2) = 8^(3-2) = 8
Also, (adj A^2)^(-1) = (adj A)^2 (since adj A is invertible)
So, adj(adj A^2) = (adj A)^2 |adj A^2| = (2A)^2 * 8 = 32A^2
Finally, let's calculate |adj(adj A^2)|.
|adj(adj A^2)| = |32A^2| = 32^n |A^2| = 32^3 * 8 = 7168
Therefore, the answer is option (d) None of these, since none of the given options match our calculated value of |adj(adj A^2)|.
Given, |A| = 2
We know that adj(adj A) = |A|^(n-2) A, where n is the order of matrix A.
Here, n = 3 (since A is a 3-rowed square matrix)
So, adj(adj A) = 2^(3-2) A = 2A
Now, let's calculate A^2.
A^2 = A.A
We know that |A.A| = |A|.|A| = |A|^2
So, |A.A| = 2^2 = 4
Now, let's calculate adj(A^2).
adj(A^2) = (A^2)^(-1) |A^2|
We know that |A^2| = |A|^3 = 2^3 = 8
Also, (A^2)^(-1) = (A^(-1))^2 (since A is invertible)
So, adj(A^2) = (A^(-1))^2 |A^2| = 8(A^(-1))^2
Now, let's calculate adj(adj A^2).
adj(adj A^2) = (adj A^2)^(-1) |adj A^2|
We know that |adj A^2| = |A^2|^(n-2) = 8^(3-2) = 8
Also, (adj A^2)^(-1) = (adj A)^2 (since adj A is invertible)
So, adj(adj A^2) = (adj A)^2 |adj A^2| = (2A)^2 * 8 = 32A^2
Finally, let's calculate |adj(adj A^2)|.
|adj(adj A^2)| = |32A^2| = 32^n |A^2| = 32^3 * 8 = 7168
Therefore, the answer is option (d) None of these, since none of the given options match our calculated value of |adj(adj A^2)|.
Community Answer
If A is a 3-rowed square matrix such that |A| =2, then then|adj(adj(ad...
|adj(A)|=|A|^(n-1) |adj(adjA)|=|A|^((n-1)^2) |adj(adjA^2)| = [|A^2|^((n-1)^2)]^2 here n order=3 , |adj(adjA^2)| = (2^2^(3-1)^2)^2 =2^16
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If A is a 3-rowed square matrix such that |A| =2, then then|adj(adj(adjA))|is?a)512b)64c)256d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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