Let A = pmatrix 1 & & 1 - & 1 & -1 & - & 1 pmatrix , where R . The maximum value of | adj (2A) | is equal to

Step-by-step solution

The given matrix is A = pmatrix 1 & & 1 - & 1 & -1 & - & 1 pmatrix . Expanding the determinant of A along the first row: |A| = 1(1 - (- ^2 )) - (- - (- )) + 1( ^2 - (-1)) |A| = 1(1 + ^2 ) - (0) + 1( ^2 + 1) |A| = 2(1 + ^2 ) Since 0 ^2 1 for real , the maximum value of |A| is 2(1 + 1) = 4 . We need to find the maximum value of | adj (2A)| . For a 3 3 matrix X , | adj (X)| = |X|^2 and |kX| = k^3 |X| . Using these properties: | adj (2A)| = |2A|^2 |2A| = 2^3 |A| = 8|A| Therefore, | adj (2A)| = (8|A|)^2 = 64|A|^2 Substi