Class 12 Mathematics (041) · Chapter 4 NotesDeterminants
Comprehensive notes for Class 12 Mathematics Chapter 4: Determinants. Learn about evaluation, area of triangles, minors, cofactors, adjoints, and matrix inversion.
In this chapter, students transition from the study of matrices to the concept of determinants, which are unique numbers associated with square matrices. While a matrix is an arrangement of numbers, its determinant provides a single value (real or complex) that reveals critical properties of the matrix, such as its invertibility. The chapter explores how determinants are used to determine if a system of linear equations has a unique solution. Students will learn to evaluate determinants of order one, two, and three, and understand the roles of minors and cofactors in these calculations. Beyond basic evaluation, the chapter covers practical applications including finding the area of a triangle using coordinates, calculating the adjoint and inverse of a matrix, and employing the Matrix Method to solve systems of linear equations in two or three variables. These concepts are foundational for advanced studies in science, engineering, and economics, where linear modeling is essential.
What you'll learn
1Define a determinant as a number associated with every square matrix.
2Evaluate determinants of order 1, 2, and 3 using row or column expansion.
3Calculate the area of a triangle and verify collinearity of points using determinants.
4Identify and compute minors and cofactors for elements of a square matrix.
5Determine the adjoint of a matrix and use it to find the inverse of non-singular matrices.
6Solve systems of linear equations using the matrix method (X = A⁻¹B).
7Distinguish between consistent and inconsistent systems of equations based on solution existence.
Chapter at a glance
01Determinant of a Square Matrix
02Properties of Determinants
03Adjoint and Inverse of a Matrix
04Applications of Determinants and Matrices
Detailed chapter notes
01
Understanding the Determinant
A determinant is a scalar value associated with a square matrix. It is denoted by |A|, det A, or the Greek letter Δ (delta). It is important to note that |A| represents the determinant of A and not its absolute or modulus value. Only square matrices possess determinants. For a matrix of order 1, the determinant is the element itself. For a matrix of order 2, the determinant is calculated by the difference of the products of the diagonal elements. For order 3, the determinant is evaluated by expanding it along any one of its three rows or three columns, resulting in the same unique value regardless of the path chosen.
Order 1If A = [a], then det A = a.
Order 2If A = [[a, b], [c, d]], then det A = ad - bc.
Order 3Evaluated by expressing it in terms of second-order determinants.
02
Expansion of Third Order Determinants
To evaluate a 3x3 determinant, we use the method of expansion along a row or column. This involves multiplying each element of the chosen row or column by its corresponding cofactor. For easier calculation, it is often best to expand along the row or column containing the most zeros. When expanding, each element is multiplied by (-1) raised to the power of the sum of its row and column indices, then multiplied by the 2x2 determinant remaining after deleting that element's row and column.
There are six ways to expand a 3x3 determinant (3 rows and 3 columns).
The sign of an element in expansion is determined by (-1)^(i+j).
If A = kB for matrices of order n, then |A| = kⁿ|B|.
03
Area of a Triangle and Collinearity
Determinants provide a streamlined formula for calculating the area of a triangle when the coordinates of its vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) are known. The area is half the absolute value of the determinant formed by the coordinates and a column of ones. Since area must be positive, the absolute value is always taken. If the resulting area is zero, it signifies that the three points do not form a triangle and are, therefore, collinear. This application also allows for finding the equation of a line passing through two points by setting the determinant to zero with a variable point (x, y).
Collinear points result in a determinant value of zero.
When area is given, use both positive and negative values for calculations.
04
Minors and Cofactors
Minors and cofactors are the building blocks for finding the adjoint and inverse of a matrix. A minor (Mᵢⱼ) of an element is the determinant obtained by deleting the row and column in which that element lies. A cofactor (Aᵢⱼ) is the minor multiplied by a positional sign, defined as (-1) raised to the power of the sum of the row and column indices. A key property is that the sum of the products of elements of any row with their corresponding cofactors equals the determinant value, while the sum of products of elements of one row with cofactors of another row is always zero.
Minor MᵢⱼDeterminant of order n-1 for a matrix of order n.
Cofactor Aᵢⱼ = (-1)^(i+j) Mᵢⱼ.
Determinant Δ = a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃.
05
Adjoint and Inverse of a Matrix
The adjoint of a square matrix A is the transpose of the matrix of its cofactors. It is denoted as adj A. A fundamental theorem states that A(adj A) = (adj A)A = |A|I, where I is the identity matrix. This relationship leads to the definition of the inverse matrix. A matrix A is invertible if and only if it is non-singular, meaning its determinant is not zero. The inverse A⁻¹ is calculated by dividing the adjoint of A by its determinant. Singular matrices, where the determinant is zero, do not have an inverse.
AdjointTranspose of the cofactor matrix [Aᵢⱼ].
Non-singular matrix|A| ≠ 0.
Inverse FormulaA⁻¹ = (1/|A|) adj A.
Property|adj A| = |A|^(n-1) for a matrix of order n.
06
Solving Systems of Linear Equations
Determinants and matrices are used to solve systems of linear equations through the Matrix Method. A system can be represented as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix. If A is non-singular, the unique solution is found using X = A⁻¹B. The system is called consistent if at least one solution exists and inconsistent if no solution exists. For singular matrices where |A| = 0, the system is inconsistent if (adj A)B is not zero. If (adj A)B is zero, the system may have infinitely many solutions or no solution.
Consistent systemSolution exists.
Inconsistent systemNo solution exists.
Matrix MethodX = A⁻¹B provides a unique solution when |A| ≠ 0.
Want the complete chapter resources?Topic notes, quizzes and flashcards for Determinants.
What is the determinant of the matrix [[3, 0], [0, 3]]?
A9
B6
C3
D0
Show answer
Answer: (A) 9
For a 2×2 diagonal matrix [[a, 0], [0, d]], the determinant is a×d. Here, 3×3 = 9.
Question 02
If a determinant has two identical rows, what is its value?
AZero
BOne
CNegative
DCannot be determined
Show answer
Answer: (A) Zero
By the property of determinants, if any two rows (or columns) are identical, the determinant equals zero.
Question 03
For a 3×3 matrix, if M₁₂ denotes the minor of element a₁₂, what is M₁₂?
AThe determinant of the 2×2 matrix obtained by deleting row 1 and column 2
BThe determinant of the 2×2 matrix obtained by deleting row 2 and column 1
CThe cofactor of element a₁₂
DThe product of elements in row 1 and column 2
Show answer
Answer: (A) The determinant of the 2×2 matrix obtained by deleting row 1 and column 2
The minor M₁₂ is obtained by deleting row 1 and column 2 from the original matrix and computing the determinant of the resulting 2×2 matrix. This is the fundamental definition of a minor.
Question 04
For a square matrix A of order n, if adj(A) denotes the adjoint of A, then what is the relationship between A and adj(A)?
AA · adj(A) = |A| · I
BA · adj(A) = A · I
CA + adj(A) = I
DA · adj(A) = 0
Show answer
Answer: (A) A · adj(A) = |A| · I
The fundamental property of adjoint matrix states that A · adj(A) = adj(A) · A = |A| · I, where |A| is the determinant of A and I is the identity matrix.
Question 05
If A is a 2×2 matrix and |A| = 5, then |3A| equals:
A15
B45
C5
D25
Show answer
Answer: (B) 45
For a 2×2 matrix, |kA| = k²|A|. Therefore, |3A| = 3² × 5 = 9 × 5 = 45.
Ready for more practice?Unlock the full quiz for this chapter.
What is the difference between a matrix and a determinant?
A matrix is an ordered rectangular array of numbers or functions and does not have a single numerical value. In contrast, a determinant is a unique number associated only with a square matrix, calculated using specific algebraic rules.
Why can only square matrices have determinants?
Determinants are defined based on the concept of associating a unique number to a system of linear equations where the number of variables equals the number of equations. This mathematical structure naturally corresponds only to square matrices.
How do you identify a singular matrix?
A square matrix is identified as singular if the value of its determinant is exactly zero (|A| = 0). Singular matrices are important because they do not possess a multiplicative inverse.
What does it mean if the determinant of a triangle's vertices is zero?
If the determinant calculated using the coordinates of three points is zero, it means the area of the triangle formed by them is zero. This proves that the three points are collinear, meaning they lie on the same straight line.
What is the Matrix Method for solving equations?
The Matrix Method is a technique to solve a system of linear equations AX = B by finding the inverse of the coefficient matrix A. If A is non-singular, the variables are found using the formula X = A⁻¹B.