This chapter covers determinants, their properties, and applications, which are essential for solving linear equations using matrices.
Determinants – Formula & Equation Sheet
Essential formulas and equations from Mathematics Part - I, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Determinants chapter of Mathematics Part - I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
|A| = a11 a22 - a12 a21
For a 2x2 matrix A = [[a11, a12], [a21, a22]], |A| represents the determinant. It measures the scaling factor of the area when the matrix transforms a space.
|A| = a11 (a22 a33 - a23 a32) - a12 (a21 a33 - a23 a31) + a13 (a21 a32 - a22 a31)
For a 3x3 matrix A, this formula calculates the determinant by expanding along a row. Useful in finding the volume of the parallelepiped defined by the column vectors.
Area of triangle = 1/2 * |A|
Given vertices (x1, y1), (x2, y2), (x3, y3), the area of the triangle can also be expressed as the absolute value of the determinant of matrix formed by these coordinates.
If A = kB, |A| = k^n |B|
For square matrices A and B of order n, if A is a scalar multiple of B, the determinant scales by k raised to the order of matrix.
|adj A| = |A|^(n-1)
For an n x n matrix A, the determinant of its adjoint is equal to the determinant of A raised to the power of (n-1).
Cofactor Aij = (-1)^(i+j) Mij
The cofactor of the element a_ij is defined as (-1) raised to the sum of its row and column indices multiplied by its minor M_ij, which is the determinant of the matrix after removing its row and column.
|A| = sum(a_ij Aij for i-th row)
The determinant can be computed by taking the sum of the products of elements of any row (or column) with their corresponding cofactors.
If |A| = 0, A is singular.
A matrix is singular if its determinant equals zero, indicating that it does not have an inverse.
|AB| = |A| |B|
The determinant of the product of two matrices is equal to the product of their determinants, a property that holds for square matrices.
Area = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
This formula calculates the signed area of a triangle in Cartesian coordinates defined by points (x1, y1), (x2, y2), (x3, y3). Always take the absolute value for positive area.
Equations
det(A) = |A|
This defines the notation; the determinant of matrix A is denoted as |A|.
Row operations do not change the determinant.
Swapping rows or adding a multiple of one row to another does not change the value of the determinant.
If a row (or column) is a linear combination of others, |A| = 0.
If the rows (or columns) of a matrix are not linearly independent, its determinant is zero.
|A| for order 3 expanded along R1 = a11 * C1 + a12 * C2 + a13 * C3
This represents a general expansion of a determinant of a 3x3 matrix along the first row.
|A| = 0 = d1 det(A1) + d2 det(A2) + ... + dn det(An)
This equation holds for a system of equations and indicates that if the determinants of the submatrices are zero, the whole determinant is zero.
A_ij = (-1)^(i+j) M_ij
Defines the cofactor associated with the matrix element a_ij.
x = A^{-1}B
In a system of linear equations represented as AX = B, the solution X can be found using the inverse of matrix A, provided it exists.
det(kA) = k^n * det(A)
For any scalar k and an n x n matrix A, multiplying A by k multiplies its determinant by k raised to the order of the matrix.
Area of triangle = 1/2 * |det|
Unaffected by the order of vertices, the formula remains the same.
|adj A| = |A|^(n-1)
If A is an n x n matrix, the determinant of its adjoint adj A is equal to |A| raised to (n-1).
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