SUMMARY: The chapter on Determinants in Class 12 Mathematics explores the properties, applications, and methods of calculating determinants of matrices.
KEY TOPICS: definition of determinants, properties of determinants, minors and cofactors, applications of determinants, Cramer's Rule, adjoint and inverse of a matrix, area of a triangle using determinants, consistency of a system of linear equations, determinant of a matrix of order 3.
The value of |A| where A = [[2, 3], [4, 5]] is:
For an invertible matrix A of order n, |adj A| equals:
If A is a 3×3 matrix with |A| = 5, then |adj A| equals:
The system of equations 2x + 3y = 5 and 4x + 6y = 10 has:
The area of the triangle with vertices A(1, 2), B(3, 4), C(5, 6) using determinants equals:
Find |A| if A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Find x such that |[[2, x], [3, 4]]| = 0.
State Cramer's rule for a 2 × 2 system of equations.
Find the area of the triangle with vertices (1, 0), (4, 0), (4, 4) using determinants.
Show that if A is a square matrix and |A| = 0, then A is singular.
Using cofactor expansion, evaluate |A| where A = [[1, −1, 2], [3, 0, 1], [−1, 2, 4]].
Solve by Cramer's rule: 2x + y − z = 3; x − y + z = 0; 3x + y + 2z = 5.
Find the area of the triangle whose vertices are A(2, 7), B(1, 1), C(10, 8) using determinants.
For the system: x + y + z = 6; x − y + z = 2; 2x + y − z = 1, find values of x, y, z using matrix method.
Show that the determinant |[[a, a², a³ − 1], [b, b², b³ − 1], [c, c², c³ − 1]]| = (a − b)(b − c)(c − a)(abc − 1).
Compare singular and non-singular matrices with the help of a table.
Assertion (A): For a 3×3 invertible matrix A: |adj A| = |A|².
Reason (R): Taking the determinant of A · adj A = |A| I gives |A| · |adj A| = |A|³.
Assertion (A): Three points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear if and only if the determinant of their coordinate matrix is zero.
Reason (R): Collinear points produce a degenerate triangle of zero area, and area is half the absolute value of that determinant.
Assertion (A): Cramer's rule applies only when the determinant of the coefficient matrix is non-zero.
Reason (R): A zero determinant means the system has either no solution or infinitely many solutions.
Assertion (A): A singular matrix has no multiplicative inverse.
Reason (R): Its determinant is zero.
Assertion (A): The determinant of a square matrix can be computed by cofactor expansion along any row or column.
Reason (R): All cofactor expansions yield the same scalar value because of multilinearity.
Statement 1: Interchanging two rows changes the sign of the determinant.
Statement 2: Adding a multiple of one row to another does not change the determinant.
Statement 1: The determinant of a triangular matrix is the product of its diagonal entries.
Statement 2: Cofactor expansion along the column with zeros yields the diagonal product directly.
Statement 1: If a row of a determinant has all entries zero then the determinant is zero.
Statement 2: A determinant equals the sum of products along the zero row.
Statement 1: A system of linear equations has a unique solution iff the determinant of the coefficient matrix is non-zero.
Statement 2: Unique solution requires invertibility of the coefficient matrix.
Statement 1: Three points are collinear iff the determinant formed by their coordinates is zero.
Statement 2: Zero area for the triangle formed by the points implies collinearity.
Study the determinant properties of the following matrices:
| Matrix | Determinant | Singular? |
|---|---|---|
| A = [[1,2],[3,4]] | −2 | No |
| B = [[2,4],[1,2]] | 0 | Yes |
| C = [[5,0],[0,3]] | 15 | No |
| D = [[2,3],[6,9]] | 0 | Yes |
| I = [[1,0],[0,1]] | 1 | No |
Study the determinant properties under operations:
| Operation | Effect on determinant |
|---|---|
| Interchange two rows | Sign of det reverses |
| Multiply a row by k | det is multiplied by k |
| Add a multiple of one row to another | det unchanged |
| All elements of a row are zero | det = 0 |
| Two rows identical | det = 0 |
For the matrix A = [[3, 1, 2], [0, 4, −1], [2, 0, 1]], compute (i) the cofactor of the entry a₂₂ = 4, (ii) the determinant of A by expansion along R1.
| Row | Entries |
|---|---|
| R1 | [3, 1, 2] |
| R2 | [0, 4, −1] |
| R3 | [2, 0, 1] |
Solve the following system of equations using Cramer's rule: 2x + y − z = 3, x + 2y + z = 5, 3x − y + 2z = 4. Compute D, Dx, Dy, Dz.
| Equation | Form |
|---|---|
| E1 | 2x + y − z = 3 |
| E2 | x + 2y + z = 5 |
| E3 | 3x − y + 2z = 4 |
Study the triangle ABC plotted from its vertices and answer:
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