This chapter covers determinants, their properties, and applications, which are essential for solving linear equations using matrices.
Determinants - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics Part - I.
This compact guide covers 20 must-know concepts from Determinants aligned with Class 12 preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Determinant of a matrix A.
A determinant is a scalar value computed from a square matrix A, indicating certain properties like invertibility.
Determinant of a 1x1 matrix.
For A = [a], det(A) = a. It's simply the element itself.
Determinant of a 2x2 matrix.
For A = [[a₁₁, a₁₂], [a₂₁, a₂₂]], det(A) = a₁₁ * a₂₂ - a₂₁ * a₁₂.
Determinant of a 3x3 matrix.
For A = [[a₁₁, a₁₂, a₁₃], [a₂₁, a₂₂, a₂₃], [a₃₁, a₃₂, a₃₃]], expand along any row/column.
Expansion using minors and cofactors.
det(A) can be computed using elements of a row multiplied by their respective cofactors.
Area of a triangle using determinants.
Area = 1/2 * |det([[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]])| gives the triangle’s area.
Properties of determinants.
If two rows/columns are interchanged, the determinant changes sign. If identical, the determinant is zero.
Singular and non-singular matrices.
A matrix is singular if its determinant is zero; otherwise, it is non-singular (invertible).
If A = kB, then det(A) = k^n * det(B).
This holds for square matrices of order n, where k is a scalar.
Cramer's Rule.
It provides a way to solve linear equations using determinants: x = det(Aₓ)/det(A), etc.
Inverse of a matrix using adjoints.
If A is non-singular, A⁻¹ = (1/det(A)) * adj(A), wherein adj(A) is the transpose of cofactor matrix.
Adjoint of a matrix.
The adjoint of A consists of the cofactors and is used in finding the inverse of A.
Consistency of linear equations.
A system of equations is consistent if det(A) ≠ 0 indicating a unique solution.
Linear combinations and determinants.
If two rows/columns can be written as combinations of others, det(A) = 0.
Properties involving row transformations.
Multiplying a row by a constant multiplies the determinant by that constant.
Effect of adding rows.
Adding a multiple of one row to another does not change the determinant.
Determinants of a triangular matrix.
For triangular matrices, the determinant is the product of the diagonal elements.
Characteristic polynomial and eigenvalues.
Eigenvalues can be found by resolving det(A - λI) = 0, where λ is an eigenvalue.
Determinant notation.
det(A) is often denoted by |A|, where |A| indicates its determinant.
3D volume using determinants.
Volume of a parallelepiped formed by vectors can be calculated by the determinant of a matrix formed by the vectors.
Inverse exists if det(A) ≠ 0.
For any square matrix, if the determinant is non-zero, the matrix is invertible.
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