CBSE Class 12 Maths Chapter 4: Determinants Revision Notes

Key Definitions

Determinant

A scalar value associated with every square matrix. It is denoted by det(A) or |A|. For a 2x2 matrix A = [[a, b], [c, d]], det(A) = ad - bc.

Minor (Mij)

The determinant of the submatrix obtained by deleting the i-th row and j-th column of a square matrix. It's a determinant of a matrix of order (n-1).

Cofactor (Aij)

Defined as Aij = (-1)^(i+j) * Mij, where Mij is the minor of the element aij. It incorporates the sign along with the minor.

Adjoint of a Matrix (adj(A))

The transpose of the cofactor matrix. If C is the cofactor matrix, then adj(A) = C^T. It's essential for finding the inverse.

Singular Matrix

A square matrix A is called singular if its determinant is zero, i.e., det(A) = 0. Singular matrices do not have an inverse.

Non-Singular Matrix

A square matrix A is called non-singular if its determinant is non-zero, i.e., det(A) ≠ 0. Non-singular matrices always possess an inverse.

Inverse of a Matrix (A⁻¹)

For a non-singular square matrix A, its inverse A⁻¹ exists such that A A⁻¹ = A⁻¹ A = I, where I is the identity matrix. It is given by the formula A⁻¹ = (1/det(A)) * adj(A).

Evaluating Determinants and Their Properties

Understanding how to evaluate determinants is fundamental. For a 2x2 matrix A = [[a, b], [c, d]], the determinant det(A) is simply ad - bc. For a 3x3 matrix, the determinant can be calculated by expanding along any row or column using cofactors. For example, expanding along the first row: det(A) = a₁₁(A₁₁) + a₁₂(A₁₂) + a₁₃(A₁₃), where A₁₁, A₁₂, A₁₃ are the cofactors of the elements a₁₁, a₁₂, a₁₃ respectively. Always remember to use the correct sign (-1)^(i+j) for cofactors.

Properties of Determinants are crucial for simplifying calculations and proving various results without direct expansion:

1. Row/Column Interchange: If any two rows (or columns) of a determinant are interchanged, the sign of the determinant changes.
2. Identical Rows/Columns: If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
3. Scalar Multiplication: If each element of a row (or column) of a determinant is multiplied by a constant k, then its value gets multiplied by k. This implies det(kA) = k^n det(A) for an n x n matrix A.
4. Row/Column Operations: If to any row (or column) of a determinant, a multiple of another row (or column) is added, the value of the determinant remains the same. (e.g., Rᵢ → Rᵢ + kRⱼ or Cᵢ → Cᵢ + kCⱼ).
5. Zero Row/Column: If all elements of a row (or column) are zero, the determinant is zero.
6. Sum Property: If elements of a row (or column) are expressed as a sum of two or more terms, the determinant can be expressed as a sum of two or more determinants.
7. det(AB) = det(A)det(B): The determinant of the product of two matrices is the product of their determinants.

Key Formulas and Must-Remember Points

• Determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc.
• Cofactor Aij = (-1)^(i+j) * Mij.
• Adjoint of A, adj(A), is the transpose of the cofactor matrix of A.
• Inverse of a matrix A⁻¹ = (1/det(A)) * adj(A), provided det(A) ≠ 0.
• A matrix A is singular if det(A) = 0; it is non-singular if det(A) ≠ 0.
• A adj(A) = adj(A) A = det(A) * I (where I is the identity matrix).
• Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|, which can be written as (1/2) |det([[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]])|.
• If three points are collinear, the area of the triangle formed by them is zero, hence det([[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]]) = 0.
• Equation of a line passing through two points (x₁, y₁) and (x₂, y₂) using determinants: det([[x, y, 1], [x₁, y₁, 1], [x₂, y₂, 1]]) = 0.
• det(AB) = det(A)det(B) and det(Aᵀ) = det(A).

Worked Examples

• {"title":"Example 1: Calculate a 3x3 Determinant","bodyMarkdown":"Find the determinant of A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].\n\nSolution: Expanding along the first row:\ndet(A) = 1 det([[5, 6], [8, 9]]) - 2 det([[4, 6], [7, 9]]) + 3 det([[4, 5], [7, 8]])\ndet(A) = 1 (59 - 68) - 2 (49 - 67) + 3 (48 - 57)\ndet(A) = 1 (45 - 48) - 2 (36 - 42) + 3 (32 - 35)\ndet(A) = 1 (-3) - 2 (-6) + 3 (-3)\ndet(A) = -3 + 12 - 9 = 0"}
• {"title":"Example 2: Find Adjoint of a 2x2 Matrix","bodyMarkdown":"Find adj(A) for A = [[2, 3], [1, 4]].\n\nSolution:\nCofactors:\nA₁₁ = (-1)^(1+1) M₁₁ = 1 4 = 4\nA₁₂ = (-1)^(1+2) M₁₂ = -1 1 = -1\nA₂₁ = (-1)^(2+1) M₂₁ = -1 3 = -3\nA₂₂ = (-1)^(2+2) M₂₂ = 1 2 = 2\n\nCofactor matrix C = [[4, -1], [-3, 2]]\nadj(A) = Cᵀ = [[4, -3], [-1, 2]]"}

Steps to Find the Inverse of a Matrix (A⁻¹)

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Exam Tip: Avoiding Common Mistakes

When solving determinant problems, always be careful with signs while calculating cofactors ((-1)^(i+j) is a common trap). For property-based questions, explicitly state the property used for each step, as this often carries marks. Remember that the area of a triangle calculated using determinants must always be a positive value, so take the absolute value. If a problem asks to prove collinearity, show that the determinant for the area of the triangle is zero. Lastly, for finding inverse, double-check your determinant calculation – a single error here invalidates the entire inverse.

Practice Questions with Solutions

• Q: What is the condition for a square matrix to be invertible? A: A square matrix is invertible if and only if it is non-singular, meaning its determinant is non-zero (det(A) ≠ 0).
• Q: How does det(kA) relate to det(A) for an n x n matrix A? A: det(kA) = k^n * det(A). The scalar k is raised to the power of the matrix's order n.
• Q: If adj(A) = [[2, -1], [3, 4]] and det(A) = 5, what is A⁻¹? A: A⁻¹ = (1/det(A)) adj(A) = (1/5) [[2, -1], [3, 4]] = [[2/5, -1/5], [3/5, 4/5]].
• Q: If two rows of a determinant are identical, what is its value? A: If any two rows (or columns) of a determinant are identical, its value is zero. This is a key property of determinants.

Frequently Asked Questions