If \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\)\(\begin{pmatrix} 5 \\ 4 \end{pmatrix}\) = k\(\begin{pmatrix} 17.5 \\ 40.0 \end{pmatrix}\), find the valu...

Answer Details

To find the value of \(k\), we need to calculate the product of the given matrices and compare it with the given result. Multiplying the first matrix by the second matrix, we get: \(\begin{pmatrix}  2  &  1 \\  4 & 3 \end{pmatrix}\)\(\begin{pmatrix}  5 \\ 4 \end{pmatrix}\) = \(\begin{pmatrix}  2(5)+1(4)  \\  4(5)+3(4) \end{pmatrix}\) = \(\begin{pmatrix}  14  \\  32 \end{pmatrix}\) Now, we can compare this result with the given result: k\(\begin{pmatrix}  17.5 \\ 40.0 \end{pmatrix}\) We see that the first element of the calculated result, 14, is less than the first element of the given result, 17.5. This means that k must be less than 1. Dividing both sides of the given equation by \(\begin{pmatrix}  17.5 \\ 40.0 \end{pmatrix}\), we get: \(\begin{pmatrix}  2  &  1 \\  4 & 3 \end{pmatrix}\)\(\begin{pmatrix}  5 \\ 4 \end{pmatrix}\) / \(\begin{pmatrix}  17.5 \\ 40.0 \end{pmatrix}\) = k Calculating the left-hand side, we get: \(\begin{pmatrix}  2  &  1 \\  4 & 3 \end{pmatrix}\)\(\begin{pmatrix}  5 \\ 4 \end{pmatrix}\) / \(\begin{pmatrix}  17.5 \\ 40.0 \end{pmatrix}\) = \(\begin{pmatrix}  14/35  \\  32/40 \end{pmatrix}\) = \(\begin{pmatrix}  0.4  \\  0.8 \end{pmatrix}\) Therefore, the value of \(k\) is equal to the first element of this result, which is 0.4. Hence, the correct answer is option (C) 0.8.