The lecture derives Cramer's rule by interpreting coordinates as signed areas or volumes and using how a determinant scales them under a linear transformation.
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A linear system can be viewed as a matrix transformation: the unknown input vector is mapped to a known output. When the determinant is nonzero, each output has exactly one input. Dot products might seem like a way to recover the input’s coordinates, but most transformations do not preserve dot products. The lecture instead interprets coordinates geometrically: in two dimensions, a coordinate can be represented by the signed area of a parallelogram formed from the vector and a basis vector. In three dimensions, the analogous quantity is a signed volume.
A transformation scales all areas or volumes by the same factor: the determinant of its matrix. Thus, the relevant signed area or volume after transformation can be calculated from the known output and the matrix columns, then divided by the determinant to recover the original coordinate. This gives Cramer’s rule: replace the appropriate column of the coefficient matrix with the output vector, take the resulting determinant, and divide by the determinant of the original matrix. The same reasoning extends to larger systems. Cramer’s rule is not presented as the fastest computational method; its value is showing how determinants and solutions to linear systems are connected.
Correct answer: A. A nonzero determinant means the transformation is invertible, so each possible output corresponds to exactly one input.
Correct answer: D. Since the columns are orthonormal, each column has dot product 1 with itself and 0 with every other column. Thus, dotting b with column i isolates xᵢ.
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