If the transpose of matrix A is matrix B, where \(\mathrm{A} = \begin{bmatrix} 2a & 1 & 3b \\ 1 & 2 & 4c \\ 5 & 6 & 0 \end{bmatrix}\) and \(\mathrm{B} = \begin{bmatrix} 4 & 1 & 5 \\ 1 & 2 & 6 \\ 9 & 3 & 0 \end{bmatrix}\) then the value of \(3a + 2b + 4c\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 15
The transpose turns the rows of A into columns:
\( \displaystyle A^{T} = \begin{bmatrix} 2a & 1 & 5 \\ 1 & 2 & 6 \\ 3b & 4c & 0 \end{bmatrix} \)
By equality of matrices, \(A^{T} = B\) means the corresponding entries are equal:
\(2a = 4 \Rightarrow a = 2\)
\(3b = 9 \Rightarrow b = 3\)
\( \displaystyle 4c = 3 \Rightarrow c = \frac{3}{4} \)
So
\( \displaystyle 3a + 2b + 4c = 6 + 6 + 4 \times \frac{3}{4} = 15 \)