Determinants Matrices Question 101
Question: If $ B = \begin{bmatrix} 3 & -1 & 2 \\ 1 & 4 & -3 \\ 2 & 0 & 5 \end{bmatrix} $, find the determinant of the adjoint of the adjoint of matrix B.
Options:
A) $ 14^4 $
B) $ 14^3 $
C) $ 14^2 $
D) $ 14^1 $
Show Answer
Answer:
Correct Answer: A
Solution:
Let’s first calculate the adjoint of matrix B:
Calculate the cofactor matrix of B: $ B_{11} = \begin{vmatrix} 4 & -3 \\ 0 & 5 \end{vmatrix} = 20 $
$ B_{12} = -\begin{vmatrix} 1 & -3 \\ 2 & 5 \end{vmatrix} = -7 $
$ B_{13} = \begin{vmatrix} 1 & 4 \\ 2 & 0 \end{vmatrix} = -8 $
$ B_{21} = -\begin{vmatrix} -1 & 2 \\ 0 & 5 \end{vmatrix} = 5 $
$ B_{22} = \begin{vmatrix} 3 & 2 \\ 2 & 5 \end{vmatrix} = 11 $
$ B_{23} = -\begin{vmatrix} 3 & -1 \\ 2 & 0 \end{vmatrix} = 6 $
$ B_{31} = \begin{vmatrix} -1 & 4 \\ 4 & -3 \end{vmatrix} = 13 $
$ B_{32} = -\begin{vmatrix} 3 & 2 \\ 1 & -3 \end{vmatrix} = -11 $
$ B_{33} = \begin{vmatrix} 3 & -1 \\ 1 & 4 \end{vmatrix} = 13 $
Form the cofactor matrix: $ \text{adj}(B) = \begin{bmatrix} 20 & -7 & -8 \\ 5 & 11 & 6 \\ 13 & -11 & 13 \end{bmatrix} $
Now calculate the adjoint of the adjoint of B: $ \text{adj}(\text{adj}(B)) = |B|^2 \cdot B = 14^2 \cdot B = \begin{bmatrix} 196 & 0 & 0 \\ 0 & 196 & 0 \\ 0 & 0 & 196 \end{bmatrix} $
BETA
