Skip to content
← Mathematics and Statistics

Mathematics and Statistics · Class 11 Commerce

Ch 6Determinants — Class 11 Mathematics and Statistics, concept-first.

Associated with every square matrix — a matrix having as many rows as columns — is a single real number that captures essential information about the matrix, above all whether a system of linear equations built from it has one definite solution. This number is the determinant of the matrix.

14

Q&A

6

Concepts

Not available

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Value of a Determinant

A determinant is a single real number computed from a square matrix. For a 2x2 matrix, a1 b1; a2 b2 = a1b2 - b1a2 (cross-multiply the diagonals and subtract).

Most relevant Q&A

Start with this concept →

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Meaning and Value of a Determinant

Associated with every square matrix — a matrix having as many rows as columns — is a single real number that captures essential information about the matrix, above all whether a system of linear equat…

2

Minors and Cofactors

Expansion by minors, used informally above, is stated precisely through two linked ideas: the minor and the cofactor of an entry.

3

Properties of Determinants

Evaluating every determinant by direct expansion is reliable but slow. A standard set of properties, all provable from the expansion rule itself, lets many determinants be simplified — or shown to be…

4

Cramer's Rule — Solving Linear Equations

Cramer's Rule uses determinants to solve a system of linear equations directly, without step-by-step elimination.

5

Consistency of a System of Linear Equations

A system of equations is consistent if it has at least one solution, and inconsistent if it has none. Determinants tell us at a glance which case a system falls into.

6

Area of a Triangle and Collinearity Using Determinants

Determinants also give a compact formula for the area of a triangle with known vertices, and — as an immediate consequence — a test for whether three points lie on one straight line.

Exercises

+Show 6 questions6 questions
  1. Q9Evaluate the determinant $\begin{vmatrix} 6 & -3 \\ 4 & 2 \end{vmatrix}$.Free
  2. Q10Evaluate $\begin{vmatrix} 2 & 0 & 1 \\ 3 & 1 & -2 \\ 1 & 4 & 5 \end{vmatrix}$ by expanding along the first row.Free
  3. Q11Find the area of the triangle whose vertices are $(2,1)$, $(4,5)$ and $(6,3)$, using determinants.Preview
  4. Q12Show, using determinants, that the points $(1,4)$, $(3,10)$ and $(-1,-2)$ are collinear.Preview
  5. Q13If a determinant of order $3$ has value $D$, and one row of it is multiplied throughout by $3$, the value of the new determinant is: (a) $D$…Preview
  6. Q14The system of equations $2x + 3y = 7$ and $4x + 6y = 5$ is: (a) consistent with a unique solution (b) consistent with infinitely many soluti…Preview

More questions

+Show 8 questions8 questions
  1. Example 1Evaluate the determinant $\begin{vmatrix} 5 & 3 \\ 2 & 4 \end{vmatrix}$.Free
  2. Example 2Evaluate $\begin{vmatrix} 1 & 2 & -1 \\ 3 & 0 & 2 \\ -2 & 1 & 4 \end{vmatrix}$ by expansion along the first row.Free
  3. Example 3Find the minors and cofactors of the elements of the second column of $A=\begin{pmatrix} 2 & -1 & 3 \\ 1 & 4 & 0 \\ 5 & 2 & 1 \end{pmatrix}$…Free
  4. Example 4Using the properties of determinants, evaluate $\begin{vmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{vmatrix}$ without full direct expan…Preview
  5. Example 5Using Cramer's Rule, solve $4x + 3y = 18$ and $2x + 5y = 16$.Preview
  6. Example 6Using Cramer's Rule, solve $x + 2y + 3z = 14$, $\ 2x - y + z = 3$, $\ 3x + y - z = 2$.Preview
  7. Example 7Examine the consistency of the system $2x + 3y = 5$ and $4x + 6y = 10$, and state the number of solutions.Preview
  8. Example 8Find the value of $k$ for which the lines $x + y - 5 = 0$, $\ x - y - 1 = 0$ and $\ 2x + ky - 12 = 0$ are concurrent.Preview