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Business Mathematics and Statistics · Class 12 Commerce

Ch 1Determinants — Class 12 Business Mathematics and Statistics, concept-first.

Every square matrix — one with the same number of rows as columns — has a single number associated with it that summarises important information about the matrix, such as whether the system of linear equations built from that matrix has a unique solution. This number is called the determinant of the matrix.

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Key concepts

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Determinant of a Square Matrix

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

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Chapter contents

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

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Meaning of a Determinant

Every square matrix — one with the same number of rows as columns — has a single number associated with it that summarises important information about the matrix, such as whether the system of linear…

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Determinants of Order Three (3×3 Determinants)

A determinant of order 3 is formed from a square matrix and is evaluated by expansion along any row or any column — every row and every column gives the same final value, which is one of the most usef…

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Minors and Cofactors

Expansion by minors, introduced informally in the previous section, is stated precisely using two related ideas: the minor and the cofactor of an entry.

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Properties of Determinants

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

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Cramer's Rule — Solving Systems of Linear Equations

Cramer's Rule uses determinants to solve a system of linear equations directly, without the row-reduction (elimination) method usually taught alongside it.

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Area of a Triangle and Collinearity of Points Using Determinants

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

Exercises

+Show 6 questions6 questions
  1. Q7Evaluate the determinant $\begin{vmatrix} 7 & -2 \\ 4 & 3 \end{vmatrix}$.Free
  2. Q8Evaluate the determinant $\begin{vmatrix} 1 & 0 & 2 \\ 3 & 1 & -1 \\ 2 & 4 & 1 \end{vmatrix}$ by expanding along the third column.Free
  3. Q9Find the area of the triangle whose vertices are $(1,2)$, $(3,4)$ and $(5,0)$, using determinants.Preview
  4. Q10Show, using determinants, that the points $(2,3)$, $(4,7)$ and $(6,11)$ are collinear.Preview
  5. Q11If two rows of a determinant are identical, the value of the determinant is: (a) $1$ (b) $-1$ (c) $0$ (d) equal to the sum of its elementsPreview
  6. Q12Using Cramer's Rule, the value of $x$ satisfying the system $2x+y=5$ and $x-y=1$ is: (a) $1$ (b) $2$ (c) $3$ (d) $-2$Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 18 questions18 questions
  1. Q1Express each of the following in one word/term each: (v) A rule with the help of which, linear equations are solved using determinants.Preview
  2. Q2(l) The number of elements, a determinant of order two contains, is (a) 2 (b) 4 (c) 6 (d) 9Preview
  3. Q3(a) Show that $\begin{pmatrix} 1 & 1 & 1 \\ 1 & -1 & 1 \\ 2 & 1 & -1 \end{pmatrix}$ is a non-singular matrix.Preview
  4. Q4If each element of a row of a determinant is multiplied by a constant 'K', then : (a) The value of the determinant remains unchanged. (b) Th…Preview
  5. Q5From the following the one which is a third order determinant, is : (a) $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$…Preview
  6. Q6Evaluate : $\begin{vmatrix} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{vmatrix}$Preview
  7. Q7Show that : $\begin{vmatrix} 1 & x & x^2 \\ 0 & y-x & y^2-x^2 \\ 0 & z-x & z^2-x^2 \end{vmatrix} = (y-x)(z-x)(x-y)$Preview
  8. Q8The co-factor of the element 8 in the following determinant is : $\begin{vmatrix} 8 & 10 \\ 12 & 14 \end{vmatrix}$ (a) 12 (b) 14 (c) $-14$ (…Preview
  9. Q9If the value of the determinant of a square matrix is not equal to zero, then the matrix is a : (a) Unit matrix (b) Zero matrix (c) Singular…Preview
  10. Q10Fill in the blanks : The value of $\begin{vmatrix} 2 & 3 \\ 0 & 5 \end{vmatrix}$ is equal to ______.Preview
  11. Q11Determine the minor and cofactor of the first element lying in first row and first column of the following determinant : $A = \begin{vmatrix…Preview
  12. Q12Find the value of the following determinant : $\begin{vmatrix} 1 & 4 & 5 \\ 3 & 6 & 9 \\ 2 & 9 & 7 \end{vmatrix}$Preview
  13. Q13Solve the following equations using Crammer's rule : $4x + 3y = 8$ $6x + 7y = 17$Preview
  14. Q14From the following determinants, the one whose value is not zero, is : (a) $\begin{vmatrix} x & 1 \\ x^2+x & x+1 \end{vmatrix}$ (b) $\begin{…Preview
  15. Q15Express each of the following in one word / term : A square matrix whose determinant is zero.Preview
  16. Q16Answer each of the following question in one sentence each : What is Cramer Rule ?Preview
  17. Q17Rectify the underlined portions of the following sentences : $\begin{vmatrix} a & d \\ b & c \end{vmatrix}$ is a _third_ order determinant.Preview
  18. Q18Write any one property of determinant.Preview

More questions

+Show 6 questions6 questions
  1. Example 1Evaluate the determinant $\begin{vmatrix} 3 & 5 \\ -1 & 2 \end{vmatrix}$.Free
  2. Example 2Evaluate the determinant $\begin{vmatrix} 2 & 3 & 1 \\ 0 & 1 & 4 \\ -1 & 2 & 0 \end{vmatrix}$ by expansion along the first row.Free
  3. Example 3Find the minors and cofactors of the elements of the second row of $A=\begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & 4 \\ 2 & 1 & 5 \end{pmatrix}$, a…Preview
  4. Example 4Using the properties of determinants, evaluate $\begin{vmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \\ 3 & 7 & 11 \end{vmatrix}$ without expanding it dir…Preview
  5. Example 5Using Cramer's Rule, solve the system of equations $3x + 2y = 12$ and $x + y = 5$.Preview
  6. Example 6Using Cramer's Rule, solve the system of equations $x+y+z=6,\ 2x-y+z=3,\ x+2y-z=2$.Preview