Answer:
a) To determine the elements of the second row of a matrix \( B' \) (transpose of matrix \( B \), assuming \( B \) is a square matrix), we need the elements from the second column of \( B \).
Let \( B = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \)
Then, \( B' = \begin{bmatrix} a & d & g \\ b & e & h \\ c & f & i \end{bmatrix} \)
So, the second row of \( B' \) is \([b, e, h]\).
b) To solve a system of linear equations using Cramer's rule, we need the matrices \( A \) and \( A_i \), where \( A_i \) is obtained by replacing the ith column of \( A \) with the column vector of constants. Without the specific matrix \( A \) provided, I can't proceed with the calculations.
c) For a matrix \( B \) to be invertible, its determinant must be non-zero. So, find the values of \( m \) such that \( \text{det}(B) \neq 0 \). Without the specific matrix \( B \) provided, I can't calculate this determinant.
Regarding the element in the second row, third column of \( 4B^{-1} \) equaling 6, we need \( B \) to be invertible. If \( B^{-1} \) exists, then the element in the second row, third column of \( 4B^{-1} \) is \( \frac{4}{B^{-1}_{23}} \). Setting this equal to 6 and solving for \( m \) will give the required values.
