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Business Mathematics and Statistics · Class 11 Commerce

Ch 1Matrices and Determinants — Class 11 Business Mathematics and Statistics, concept-first.

In business, data such as sales figures across branches, costs of raw materials, or production quantities across time periods is naturally organised into rows and columns. A matrix is exactly this: a rectangular array of numbers (called elements or entries) arranged in rows and columns, enclosed in brackets.

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Key concepts

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Matrix Types, Order and Equality

A matrix is a rectangular array of numbers arranged in rows and columns. If it has rows and columns its order is . Business data (branch-wise sales, product-wise costs) fits naturally into this shape.

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Chapter contents

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1

Matrices — Definition, Order and Types

In business, data such as sales figures across branches, costs of raw materials, or production quantities across time periods is naturally organised into rows and columns.

2

Algebra of Matrices — Addition, Subtraction, Scalar Multiplication and Transpose

Matrices of the same order can be combined by ordinary arithmetic, entry by entry.

3

Multiplication of Matrices

Matrix multiplication models a very natural business operation: combining a quantity matrix with a price/rate matrix to produce a total matrix (as in the worked revenue example below), so it is worth…

4

Determinants — Definition, Minors and Cofactors

Every square matrix has associated with it a single real number called its determinant, written or . Determinants let us test whether a system of linear equations has a unique solution, and are the ke…

5

Properties of Determinants

Expanding a determinant term by term every time is slow and error-prone. A handful of properties, applied as row/column operations, let us simplify a determinant — or spot that it is zero — often with…

6

Adjoint and Inverse of a Matrix

For a matrix , this reduces to a quick shortcut: — swap the two diagonal elements and change the sign of the two off-diagonal elements.

7

Solving a System of Linear Equations — Cramer's Rule and the Matrix Inversion Method

Matrices give two clean, systematic methods for solving simultaneous linear equations — both routinely used in business problems such as finding unit costs or allocating resources from several linear…

Exercises

+Show 7 questions7 questions
  1. Q2Using the equality of matrices $\begin{pmatrix} x+y & 2 \\ 5 & xy \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 5 & 8 \end{pmatrix}$, find the va…Free
  2. Q4Express $A=\begin{pmatrix}3&5\\1&-2\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.Free
  3. Q7If $A=\begin{pmatrix}1&0\\2&-1\end{pmatrix}$ and $B=\begin{pmatrix}3&1\\0&2\end{pmatrix}$, verify that $(AB)'=B'A'$.Free
  4. Q10For the matrix $A=\begin{pmatrix}2&-1&3\\0&4&5\\1&2&6\end{pmatrix}$, find the minor and cofactor of the elements $a_{12}$ and $a_{23}$.Preview
  5. Q12Evaluate $\begin{vmatrix}2&4&6\\1&3&5\\0&1&2\end{vmatrix}$ by first taking out a common factor from the first row.Preview
  6. Q16Solve the following system of equations using Cramer's Rule: $x+y+z=6$, $2x-y+z=3$, $x+2y-z=2$.Preview
  7. Q18A company sources a raw material from three suppliers under three different combined-purchase deals. The total costs (in Rs.\,thousands) are…Preview

Sample & Board Papers

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

+Show 29 questions29 questions
  1. Q1If $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{vmatrix}$ then $\begin{vmatrix} 3 & 1 & 2 \\ 1 & 2 & 3 \\ 2 & 3 & 1 \e…Preview
  2. Q2The value of $x$ if $\begin{vmatrix} x & x+2 \\ x-2 & x \end{vmatrix}$ is : (a) $x^2$ (b) $+4$ (c) $0$ (d) $1$Preview
  3. Q3Show that $\begin{vmatrix} x & y & z \\ 2x+2a & 2y+2b & 2z+2c \\ a & b & c \end{vmatrix} = 0$.Preview
  4. Q4Show that the matrices $A = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} \dfrac{4}{5} & -\dfr…Preview
  5. Q5If $A = \begin{bmatrix} 5 & -7 \\ 3 & -4 \end{bmatrix}$ then prove that $(A^{-1})^{-1} = A$.Preview
  6. Q6(a) Evaluate $\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix} = (a-b)(b-c)(c-a)$. OR (b) Show by the principle of ma…Preview
  7. Q7The values of $x$ if $\begin{vmatrix} 0 & 1 & 0 \\ x & 2 & x \\ 1 & 3 & x \end{vmatrix} = 0$ is : (a) $-1,\ 1$ (b) $0,\ -1$ (c) $-1,\ -1$ (d…Preview
  8. Q8The co-factor of $-7$ in the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$ is : (a) $-7$ (b) $-18$ (c) $…Preview
  9. Q9If $A = \begin{bmatrix} 2 & 4 \\ -3 & 2 \end{bmatrix}$ then, find $A^{-1}$.Preview
  10. Q10Show that the matrices $A = \begin{bmatrix} 1 & 3 & 7 \\ 4 & 2 & 3 \\ 1 & 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} \dfrac{-4}{35} & \df…Preview
  11. Q11The technology matrix of an economic system of two industries is $\begin{bmatrix} 0.6 & 0.9 \\ 0.2 & 0.8 \end{bmatrix}$. Test whether the sy…Preview
  12. Q12(a) If $A = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix}$ then, show that $(AB)^{-1} =…Preview
  13. Q13The number of Hawkin's-Simon conditions for the viability of an input-output analysis is : (a) 4 (b) 1 (c) 2 (d) 3Preview
  14. Q14If any three rows (columns) of a determinant are identical then the value of the determinant is : (a) 1 (b) 0 (c) 3 (d) 2Preview
  15. Q15Evaluate $\begin{vmatrix} x & x+1 \\ x-1 & x \end{vmatrix}$Preview
  16. Q16Find $\lambda$, if the matrix $\begin{pmatrix} 1 & 1 & 3 \\ 2 & \lambda & 4 \\ 9 & 7 & 11 \end{pmatrix}$ has no inverse.Preview
  17. Q17(a) Show that the matrices $A = \begin{pmatrix} 2 & 2 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} \frac{4}{5} & \f…Preview
  18. Q18The inventor of input-output analysis is : (a) Prof. Wassily W. Leontief (b) Sir Francis Galton (c) Arthur Caylay (d) FisherPreview
  19. Q19If $\begin{vmatrix} x & 2 \\ 8 & 5 \end{vmatrix}=0$ then the value of $x$ is : (a) $\dfrac{-16}{5}$ (b) $\dfrac{-5}{6}$ (c) $\dfrac{16}{5}$…Preview
  20. Q20Evaluate $\begin{vmatrix} 1 & 3 & 4 \\ 102 & 18 & 36 \\ 17 & 3 & 6 \end{vmatrix}$Preview
  21. Q21The technology matrix of an economic system of two industries is $\begin{bmatrix} 0.8 & 0.2 \\ 0.9 & 0.7 \end{bmatrix}$. Test whether the sy…Preview
  22. Q22(a) An economy produces only coal and steel. These two commodities serve as intermediate inputs in each other's production. 0.4 tonne of ste…Preview
  23. Q23(a) The sum of three numbers is 20. If we multiply the first by 2 and add the second number and subtract the third, we get 23. If we multipl…Preview
  24. Q24If $|A|=5$, then the value of $\left|A^{-1}\right|$ is ________. (a) $1$ (b) $5$ (c) does not exist (d) $\dfrac{1}{5}$Preview
  25. Q25The inverse matrix of $\begin{pmatrix} \dfrac{4}{5} & \dfrac{-5}{12} \\ \dfrac{-2}{5} & \dfrac{1}{2} \end{pmatrix}$ is : (a) $\dfrac{30}{7}\…Preview
  26. Q26Solve : $\begin{vmatrix} 7 & 4 & 11 \\ -3 & 5 & x \\ -x & 3 & 1 \end{vmatrix}=0$Preview
  27. Q27Solve by matrix inversion method. $2x+3y-5=0$; $x-2y+1=0$.Preview
  28. Q28(a) You are given the following transaction matrix for a two sector economy. | Sector | Sales 1 | Sales 2 | Final demand | Gross output | |…Preview
  29. Q29(a) If $A=\begin{bmatrix} 1 & 2 \\ -1 & 1 \end{bmatrix}$, $B=\begin{bmatrix} 1 & 3 \\ -1 & 2 \end{bmatrix}$ then, show that $(AB)^{-1}=B^{-1…Preview

More questions

+Show 11 questions11 questions
  1. Example 1Write the order of each of the following matrices and state whether each is a row matrix, column matrix, square matrix, diagonal matrix, sca…Free
  2. Example 3If $A=\begin{pmatrix}2&-1\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}1&5\\-2&0\end{pmatrix}$, find (i) $A+B$ (ii) $A-B$ (iii) $3A-2B$.Free
  3. Example 5If $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\1&2\end{pmatrix}$, find $AB$ and $BA$, and verify that $AB \ne BA$.Free
  4. Example 6A trader sells pens and notebooks from two shops. The quantity sold (in units) is given by $Q=\begin{pmatrix}100&50\\80&70\end{pmatrix}$, wh…Preview
  5. Example 8Evaluate the determinant $\begin{vmatrix}4&3\\2&5\end{vmatrix}$.Preview
  6. Example 9Evaluate the determinant $\begin{vmatrix}1&2&3\\0&1&4\\5&6&0\end{vmatrix}$ by expanding along the first row.Preview
  7. Example 11Without fully expanding, show that $\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}=0$.Preview
  8. Example 13Find the adjoint and the inverse of $A=\begin{pmatrix}3&1\\5&2\end{pmatrix}$.Preview
  9. Example 14Find the adjoint and inverse of $A=\begin{pmatrix}1&0&2\\0&1&1\\1&1&0\end{pmatrix}$.Preview
  10. Example 15A trader buys 3 pens and 2 pencils for Rs.\,22, and 5 pens and 4 pencils for Rs.\,38. Using Cramer's Rule, find the cost of one pen and one…Preview
  11. Example 17Solve using the matrix inversion method: $2x+3y=8$, $x-2y=-3$.Preview