Question

If \(A =[5a-b 32]\) and A.adjA=AA^T,then which of the following statements is true

• A. \(5a - b = -5\)
• B. \(det(A) < 0 \)
• C. \( 5a + b = 10\)
• D. A is symmetric
• E. \(det(A) ≥ 0\)

Answer 1

Answer: (E)

Given that:

\(adj(A) = | 27 -(5a - b) | | -32 5a |\)

and  \(A × adj(A) = A × A^T\)

Now, as pe r the question let us calculate the following:

\(A × adj(A) = | 5a - b 32 | × | 27 -(5a - b) | = | 135a - 27b - (5a - b)(32) 0 | | 32 5a | | -(32) 0 |\)

\(A × A^T = | 5a - b 32 | × | 5a - b 32 | = | (5a - b)^2 + 32^2 (5a - b)(32) | | 32 5a | | (5a - b)(32) 32^2 + 5a^2 |\)

 As per the given data \(A × adj(A) = A × A^T\)

For the (1,1) element: \(135a - 27b - (5a - b)(32) = (5a - b)^2 + 32^2\)

                                     \(    =  135a - 27b - 160a + 32b = 25a^2 - 10ab + b^2 + 1024\)

                                     \( =25a^2 - 10ab + b^2 - 27a + 5b - 1024 = 0 \)------(1)

For the (1,2) element: 0 = (5a - b)(32)

Since the determinant of a matrix is the product of its diagonal elements, we can use the (1,1) and (1,2) elements to find the determinant of A:

Therefore, \(det(A) = (5a - b) * 0 = 0\)

Now let's go through each option:

a)For, \(5a - b = -5\) We cannot conclude this from the equations we derived. So, it is not true.

b)For,  \(det(A) < 0 \)We calculated det(A) as 0, which is not less than 0. So, it is not true.

c)For, \( 5a + b = 10\) We cannot conclude this from the equations we derived. So, it is not true.

d) For, A is symmetric A matrix is symmetric if it is equal to its transpose. However, from the derived equations, we can see that A is not symmetric. So, it is not true.

e) For, \(det(A) ≥ 0\) We found that det(A) = 0. Since 0 is greater than or equal to 0, this statement is true.------ Hence this is the answer