If A= [ array cc 5 a & -b 3 & 2 array ] and A adj A=A A^t , then 5 a+b is equal to

Step-by-step solution

Given A = bmatrix 5a & -b 3 & 2 bmatrix . The property A adj A = |A| I holds for any square matrix A , where I is the identity matrix. The given equation is A adj A = A A^t . Substituting the property, we have |A| I = A A^t . Calculate the determinant |A| = (5a)(2) - (-b)(3) = 10a + 3b . Calculate A A^t : A^t = bmatrix 5a & 3 -b & 2 bmatrix A A^t = bmatrix 5a & -b 3 & 2 bmatrix bmatrix 5a & 3 -b & 2 bmatrix = bmatrix 25a^2 + b^2 & 15a - 2b 15a - 2b & 9 + 4 bmatrix = bmatrix 25a^2 + b^2 & 15a - 2b 15a - 2b & 13 bmat