1. ALGEBRA: DETERMINANTS

Introduction — What is a determinant?

A determinant is a special number associated with a square matrix. For a 2×2 or 3×3 matrix, it is a single number that tells us important information about the matrix (for example, whether the matrix is invertible). Determinants are used in solving linear equations, finding inverse matrices and in applications like area/volume and change of variables.

Determinant of 2×2 matrix (easy formula)

For matrix A = [[a, b], [c, d]], the determinant is:

det(A) = a·d − b·c

Determinant of 3×3 matrix (rule of Sarrus)

For matrix

A = [ [a, b, c], [d, e, f], [g, h, i] ]

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

(This expansion is known as expansion by first row.)

Minors and Cofactors

Minor (M_ij) is determinant of matrix that remains after removing i-th row and j-th column.

Cofactor (C_ij) = (−1)^i+j · M_ij.

Use cofactors for expansion of determinants and to compute inverse of a matrix (adjoint method).

Expansion by Cofactors (Laplace expansion)

To find determinant of an n×n matrix, pick any row (or column) and sum:

det(A) = Σ (a_ij · C_ij) over j in the chosen row (or similar over i in a column).

Tip: choose a row or column with zeros — makes calculation easier.

Properties of Determinants (Important for exams)

If two rows (or two columns) are identical → determinant = 0.
Interchanging two rows (or columns) multiplies determinant by −1.
Multiplying a row (or column) by scalar k multiplies determinant by k.
If a row (or column) is all zeros → determinant = 0.
Adding a multiple of one row to another row does not change determinant.
det(AB) = det(A) · det(B) for square matrices A and B (product rule).
det(A^T) = det(A) (transpose does not change determinant).
If A is invertible, det(A⁻¹) = 1 / det(A).

Product of Determinants

For two n×n matrices A and B,

det(AB) = det(A) × det(B)

This helps when dealing with matrix factorization — you can compute determinants of factors and multiply.

Applications of Determinants (simple examples)

Solving linear equations (Cramer’s rule): For a system of n linear equations with n unknowns, solution can be given by determinants — each unknown = determinant with replaced column divided by determinant of coefficient matrix.
Invertibility test: Matrix is invertible ⇔ its determinant ≠ 0.
Area and volume: Determinant of 2×2 matrix with column vectors gives area of parallelogram; 3×3 determinant gives volume of parallelepiped.
Change of variables: Jacobian determinant used in calculus for variable substitution (conceptual link).

Worked Examples (stepwise)

Example 1 — 2×2

Find determinant of [[3, 5], [2, 7]].

det = 3×7 − 5×2 = 21 − 10 = 11.

Example 2 — 3×3 using expansion by first row

Find determinant of

[ [1, 2, 3], [0, 4, 5], [1, 0, 6] ]

det = 1·(4·6 − 5·0) − 2·(0·6 − 5·1) + 3·(0·0 − 4·1)

= 1·24 − 2·(0 − 5) + 3·(0 − 4)

= 24 − 2·(−5) + 3·(−4) = 24 + 10 − 12 = 22.

Quick Tricks & Exam Tips

Always simplify using row/column operations before expanding — reduces calculations.
Use zeros: expand along row/column with most zeros for faster calculation.
Keep sign (−1)^i+j in mind when computing cofactors.
When a row is multiplied by a scalar, factor it out of determinant — do this early to simplify.
If determinant becomes 0 at any point (identical rows etc.) you can stop — answer is 0.

Practice Problems (for revision)

Compute determinant of [[2, 1], [5, 3]].
Compute determinant of [[1,0,2],[2,1,3],[3,4,0]] using expansion by first row.
Verify det(AB) = det(A)det(B) for A = [[1,2],[3,4]] and B = [[0,1],[1,0]].
Use Cramer’s rule to solve: x + 2y = 5, 3x − y = 4.

Key Points — Quick Revision

Determinant gives single scalar value for square matrix.
2×2 formula is a·d − b·c (memorize it).
Use minors & cofactors for larger matrices (expansion).
Row/column operations help simplify; remember how they change determinant.
determinant ≠ 0 → matrix invertible; determinant = 0 → singular.

Detailed Notes:

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