Each of the archetypes below is a system of equations with a square coefficient matrix, or is a square matrix itself. Compute the determinant of each matrix, noting how Theorem SMZD indicates when the matrix is singular or nonsingular.

Archetype A, Archetype B, Archetype F, Archetype K, Archetype L

Construct a $3\times 3$ nonsingular matrix and call it $A$. Then, for each entry of the matrix, compute the corresponding cofactor, and create a new $3\times 3$ matrix full of these cofactors by placing the cofactor of an entry in the same location as the entry it was based on. Once complete, call this matrix $C$. Compute $A\transpose{C}$. Any observations? Repeat with a new matrix, or perhaps with a $4\times 4$ matrix.

Construct an example to show that the following statement is not true for all square matrices $A$ and $B$ of the same size: $\detname{A+B}=\detname{A}+\detname{B}$.

Theorem NPNT says that if the product of square matrices $AB$ is nonsingular, then the individual matrices $A$ and $B$ are nonsingular also. Construct a new proof of this result making use of theorems about determinants of matrices.

Use Theorem DRCM to prove Theorem DZRC as a corollary. (See Proof Technique LC.)

Suppose that $A$ is a square matrix of size $n$ and $\alpha\in\complex{\null}$ is a scalar. Prove that $\detname{\alpha A}=\alpha^n\detname{A}$.

Employ Theorem DT to construct the second half of the proof of Theorem DRCM (the portion about a multiple of a column).