1. **Problem statement:** We are given the matrix $$A = \begin{bmatrix}\mathcal{a} & 1 & 1 & 1 \\ 1 & \mathcal{a} & 1 & 1 \\ 1 & 1 & \mathcal{a} & 1 \\ 1 & 1 & 1 & \mathcal{a}\end{bmatrix}$$ We need to find the determinant $\det(A)$ in terms of $\mathcal{a}$, and then determine when $A$ is singular or non-singular. 2. **Expressing the matrix:** Note that $A$ can be written as $$A = \mathcal{a}I + B$$ where $I$ is the $4\times 4$ identity matrix and $$B = \begin{bmatrix}0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0\end{bmatrix}$$ 3. **Eigenvalues approach:** The determinant depends on the eigenvalues of $A$. Since $A = \mathcal{a}I + B$, the eigenvalues of $A$ are $\mathcal{a} + \lambda_i$ where $\lambda_i$ are eigenvalues of $B$. 4. **Find eigenvalues of $B$:** Observe that $B = J - I$, where $J$ is the $4\times4$ matrix of all ones. This is because $$J = \begin{bmatrix}1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1\end{bmatrix}$$ and so $$B = J - I = \begin{bmatrix}0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0\end{bmatrix}$$ 5. **Eigenvalues of $J$:** $J$ has eigenvalues $4$ (with eigenvector $(1,1,1,1)^T$) and $0$ (with multiplicity 3). 6. **Eigenvalues of $B = J - I$:** Subtracting the identity matrix shifts eigenvalues by $-1$, so eigenvalues of $B$ are $$\lambda_1 = 4 - 1 = 3$$ (multiplicity 1) and $$\lambda_2 = 0 - 1 = -1$$ (multiplicity 3). 7. **Eigenvalues of $A$: ** $$\mu_1 = \mathcal{a} + 3$$ (multiplicity 1) $$\mu_2 = \mathcal{a} - 1$$ (multiplicity 3) 8. **Determinant using eigenvalues:** $$\det(A) = (\mathcal{a} + 3)(\mathcal{a} - 1)^3$$ 9. **Singular and non-singular:** - $A$ is singular when $\det(A) = 0$, so when either $$\mathcal{a} + 3 = 0 \Rightarrow \mathcal{a} = -3$$ or $$\mathcal{a} - 1 = 0 \Rightarrow \mathcal{a} = 1$$ - $A$ is non-singular for all other values of $\mathcal{a}$. **Final answer:** $$\det(A) = (\mathcal{a} + 3)(\mathcal{a} - 1)^3$$ The matrix $A$ is singular iff $\mathcal{a} = -3$ or $\mathcal{a} = 1$, and non-singular otherwise.