1. **Stating the problem:** We have two matrices \(X = \frac{1}{5}\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}\) and \(Y = \frac{1}{5}\begin{pmatrix}7 & a \\ b & c\end{pmatrix}\). We need to find the values of \(a\), \(b\), and \(c\) given the position of the matrix expressions is the center of the image (which implies some relation or equality between matrices). 2. **Assumption and approach:** Since the problem states "Position of the matrix expressions: center of the image" without further context, a common interpretation is that \(Y\) might be related to \(X\) by some operation such as addition, multiplication, or equality. Without additional info, the most reasonable assumption is that \(Y = X\) or that \(Y\) is a transformation of \(X\). 3. **If we assume \(Y = X\), then:** \[ \frac{1}{5}\begin{pmatrix}7 & a \\ b & c\end{pmatrix} = \frac{1}{5}\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} \] 4. **Multiply both sides by 5 to clear denominators:** \[ \begin{pmatrix}7 & a \\ b & c\end{pmatrix} = \begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} \] 5. **Equate corresponding elements:** - Top left: \(7 = 1\) (which is false, so this assumption is invalid) 6. **Alternative approach:** Perhaps \(Y\) is the product of \(X\) and some matrix or the inverse of \(X\). Let's check if \(Y = X^2\) (i.e., \(Y = X \times X\)): 7. **Calculate \(X^2\):** \[ X = \frac{1}{5}\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} \] \[ X^2 = X \times X = \left(\frac{1}{5}\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}\right)^2 = \frac{1}{25} \begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} \begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} \] Calculate the product inside: \[ \begin{pmatrix}1*1 + 2*3 & 1*2 + 2*4 \\ 3*1 + 4*3 & 3*2 + 4*4\end{pmatrix} = \begin{pmatrix}1 + 6 & 2 + 8 \\ 3 + 12 & 6 + 16\end{pmatrix} = \begin{pmatrix}7 & 10 \\ 15 & 22\end{pmatrix} \] So, \[ X^2 = \frac{1}{25} \begin{pmatrix}7 & 10 \\ 15 & 22\end{pmatrix} \] 8. **Compare \(Y = \frac{1}{5} \begin{pmatrix}7 & a \\ b & c\end{pmatrix}\) with \(X^2\):** Multiply \(Y\) by 5: \[ \begin{pmatrix}7 & a \\ b & c\end{pmatrix} = 5Y \] From \(X^2\), the matrix is \(\frac{1}{25} \begin{pmatrix}7 & 10 \\ 15 & 22\end{pmatrix}\), which is not equal to \(Y\) unless \(Y = 5X^2\). 9. **Try \(Y = 5X^2\):** Calculate \(5X^2 = 5 \times \frac{1}{25} \begin{pmatrix}7 & 10 \\ 15 & 22\end{pmatrix} = \frac{1}{5} \begin{pmatrix}7 & 10 \\ 15 & 22\end{pmatrix}\), which matches the form of \(Y\). Therefore, \[ a = 10, \quad b = 15, \quad c = 22 \] **Final answer:** \(a = 10\), \(b = 15\), \(c = 22\).