Question 1

How do you calculate a 2×2 determinant?

Accepted Answer

For A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. Multiply the main diagonal (top-left to bottom-right) and subtract the product of the anti-diagonal (top-right to bottom-left). Example: det([[3, 2], [1, 4]]) = (3)(4) − (2)(1) = 12 − 2 = 10.

Question 2

How do you expand a 3×3 determinant?

Accepted Answer

Use cofactor expansion along the first row: det(A) = a₁₁·C₁₁ + a₁₂·C₁₂ + a₁₃·C₁₃, where each Cᵢⱼ = (−1)^(i+j) × det of the 2×2 submatrix formed by deleting row i and column j. Signs alternate: + − + for the first row. You can also use the Sarrus rule: draw diagonals, sum the down-right diagonals, subtract the down-left diagonals.

Question 3

What does a zero determinant mean?

Accepted Answer

det(A) = 0 means the matrix is singular (not invertible). Geometrically, the transformation collapses the plane (for 2×2) or space (for 3×3) to a lower dimension — an area becomes a line, a volume becomes a plane. A system of equations Ax = b has no unique solution if det(A) = 0.

Question 4

What are the properties of determinants?

Accepted Answer

Key properties: det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(cA) = cⁿ·det(A) for n×n matrix; swapping two rows negates the determinant; adding a multiple of one row to another leaves the determinant unchanged; if two rows are identical, det = 0; det(I) = 1 for any identity matrix.

Question 5

What is the geometric meaning of the determinant?

Accepted Answer

The absolute value |det(A)| measures how the matrix scales areas (2×2) or volumes (3×3). If A transforms the unit square, |det(A)| is the area of the parallelogram formed by the two column vectors. If det(A) > 0 orientation is preserved; if det(A) < 0 orientation is reversed. det = 1 means area/volume is unchanged (isometry).

Question 6

How is the determinant used to find the inverse?

Accepted Answer

The inverse exists if and only if det(A) ≠ 0. For a 2×2 matrix: A⁻¹ = (1/det) × [[d, −b], [−c, a]]. For larger matrices: A⁻¹ = (1/det) × adj(A), where adj(A) is the adjugate (transpose of the cofactor matrix). Cramer's rule also uses determinants to solve Ax = b: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i with b.

Property	Formula	Meaning
Product rule	det(AB) = det(A)·det(B)	Determinant of product equals product of determinants
Transpose rule	det(Aᵀ) = det(A)	Transposing preserves the determinant
Scalar rule	det(cA) = cⁿ·det(A)	n×n matrix scaled by c — determinant scales by cⁿ
Row swap	Swap 2 rows → det changes sign	det → −det when any two rows are swapped
Identical rows	det = 0	Two identical rows make the matrix singular
Identity matrix	det(I) = 1	The identity matrix has determinant 1
Triangular matrix	det = product of diagonal	det = a₁₁·a₂₂·…·aₙₙ for triangular matrices

Matrix	a	b	c	d	det = ad−bc	Invertible?
[[1,0],[0,1]]	1	0	0	1	1	Yes (Identity)
[[3,2],[1,4]]	3	2	1	4	10	Yes
[[6,2],[3,1]]	6	2	3	1	0	No (singular)
[[2,-1],[4,3]]	2	-1	4	3	10	Yes
[[5,0],[0,5]]	5	0	0	5	25	Yes
[[0,1],[-1,0]]	0	1	-1	0	1	Yes (rotation)

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