If A = [ array cc 5 a & - b 3 & 2 array ] and A.adj A = AA ^ T , then 5 a+ b =

Step-by-step solution

Given that A adj (A) = ( A) I and A adj (A) = A A^T , it follows that ( A) I = A A^T . For matrix A = bmatrix 5a & -b 3 & 2 bmatrix , the determinant is A = (5a)(2) - (-b)(3) = 10a + 3b . Computing A A^T yields: A A^T = bmatrix 5a & -b 3 & 2 bmatrix bmatrix 5a & 3 -b & 2 bmatrix = bmatrix 25a^2 + b^2 & 15a - 2b 15a - 2b & 13 bmatrix Equating ( A) I = bmatrix 10a + 3b & 0 0 & 10a + 3b bmatrix with A A^T gives the system: 15a - 2b = 0 10a + 3b = 13 From 15a = 2b , we have b = 15 2 a . Substituting into the second equ