Question 1

How is the determinant calculated step by step?

Accepted Answer

Cofactor expansion along row 1: for each element a₁ⱼ, compute the minor matrix (delete row1, column j), then cofactor = (-1)^(1+j) * det(minor). Sum a₁ⱼ * cofactor. For 3×3, Sarrus rule is also shown.

Question 2

What does a zero determinant mean?

Accepted Answer

A zero determinant indicates a singular matrix: no unique inverse, linearly dependent rows/columns, null space dimension = n - rank. The system Ax=b may have zero or infinite solutions.

Question 3

How is the inverse matrix computed?

Accepted Answer

Using the adjugate method: A⁻¹ = adj(A) / det(A). The adjugate is the transpose of the cofactor matrix. Only for n ≤ 4 and non‑singular matrices.

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