Question

Let A be a nxn matrix such that |A| = 2. If the determinant of the matrix Adj (2.Adj(2A-¹)). is 2⁸⁴, then n is equal to___.

Answer 1

By applying properties of determinants and adjugate matrices, we find that the size of the matrix (n) for which the determinant of
`Adj (2.Adj(2A⁻¹))`is equal to
`2^(84)` is 13.

Step-by-step explanation:

We are given that the determinant of matrix A is 2 and are asked to find the size of the matrix (denoted by n) if the determinant of the matrix Adj
`(2.Adj(2A⁻¹))`is equal to
`2^(84)`. The adjugate (adjoint) of a matrix has the property where the determinant of the adjugate matrix, Adj(B), is equal to the determinant of B raised to the power of n-1. Furthermore, for any invertible matrix B, the determinant of the inverse matrix B^{-1} is 1 divided by the determinant of B.

Let's apply these rules to our given matrix, A:

1. First, consider the determinant of 2A⁻¹. Since the determinant of A is 2, the determinant of 2A⁻¹ is
   `2^n` times the determinant of
   `A^(-1),` which is 1/2, so we get
   `(2^n)*(1/2) = 2^(n-1).`
2. Next, the determinant of the adjugate of
   `2A^(-1), Adj(2A^(-1))`, will then be
   `(2^(n-1))^(n-1) = 2^((n-1)(n-1)).`
3. When we multiply by 2 to get 2.Adj(2A⁻¹), we double the determinant, yielding
   `2^((n-1)(n-1)+1).`
4. The determinant of the adjugate of this matrix is thus
   `(2^((n-1)(n-1)+1))^(n-1).`
5. This final determinant is given as 2^{84}, so we have the equation
   `(2^((n-1)(n-1)+1))^(n-1) = 2^(84).`
6. Expanding the left side gives us
   `2^((n-1)^2(n-1)+n-1)` which simplifies to
   `2^((n-1)^3) = 2^(84).`
7. Setting the exponents equal gives us
   `(n-1)^3 = 84.`
8. Solving for n, we find that n-1 =
   `2^2 * 3`, and hence
   `n = 2^2 * 3 + 1 = 13.`

Therefore, the size of the matrix n is 13.