xSkill: notes, lectures and practice for CBSE and CSE

1.1 Determinant tests and formulas A determinant is a single number attached to a square matrix. For a 2 by 2 matrix, the value is the product of the main diagonal minus the product of the other diagonal. For a 3 by 3 matrix, expand along any row or column using minors and cofactors; picking the row with a zero shortens the work. The value changes in predictable ways under row and column operations. Swapping two rows flips the sign, multiplying one row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another leaves the determinant unchanged. A matrix with two equal rows, or with a row of zeros, has determinant zero.

a c b d = a d - b c

Determinant of order two

∣ A ∣ = j = 1 \sum n a_{ij} (- 1)^{i + j} M_{ij}

Expansion along row i using minors

1.2 Area, adjoint and inverse The determinant gives the area of a triangle when the coordinates of the three vertices are placed in a 3 by 3 determinant with a third column of ones, and the result is halved. Collinearity is the special case where the area is zero. For a square matrix, each entry has a minor (the determinant left after deleting its row and column) and a cofactor (the minor with a sign from the position). The adjoint is the transpose of the cofactor matrix. The inverse exists exactly when the determinant is not zero, and it is the adjoint divided by the determinant.

Area = \frac{1}{2} x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} 111

Area of a triangle with vertices in coordinate form

A^{- 1} = \frac{1}{∣ A ∣} adj A, ∣ A ∣ \neq = 0

Inverse from the adjoint and the determinant

∣ k A ∣ = k^{n} ∣ A ∣, ∣ adj A ∣ = ∣ A ∣^{n - 1}, ∣ A^{- 1} ∣ = \frac{1}{∣ A ∣}

Scaling and adjoint determinant rules for order n

A determinant also answers whether a matrix has an inverse, and the adjoint form makes that check explicit. For an n by n matrix, the product of the matrix and its adjoint is the determinant times the identity, which is the identity used in most objective questions on this chapter.

Exam

Before using the inverse formula, check that the determinant is not zero, and write that check. A question that gives a singular matrix and asks for the inverse is testing exactly that condition.