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Business Mathematics and Statistics · Class 12 Commerce

Ch 1Applications of Matrices and Determinants (Rank, Cramer's Rule, Transition Probability Matrices) — Class 12 Business Mathematics and Statistics, concept-first.

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter extends earlier matrix and determinant work to three genuinely practical tools: finding the rank of a matrix, using rank to test whether a system of linear equations can be solved at all, solving a three-variable system by Cramer's Rule, and model…

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Key concepts

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Rank of a Matrix

The rank of a matrix is the number of non-zero rows in its row-echelon form, equivalently the order of its largest non-vanishing minor, equivalently the maximum number of linearly independent rows (or columns).

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Chapter contents

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Rank of a Matrix

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter extends earlier matrix and determinant work to three genuinely practical tools: finding the rank of a matrix, using rank to tes…

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Consistency of a System of Linear Equations — the Rank Method

For a system of linear equations in unknowns, written as , form the augmented matrix (the coefficient matrix with the constants column attached).

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Cramer's Rule for Three Variables

For the system

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Transition Probability Matrices (Markov Chains)

The transition probabilities between two states, A and B, can be read directly off the transition matrix as a from/to table:

Exercises

+Show 3 questions3 questions
  1. Q9Find the rank of $\begin{pmatrix}2&4&6\\1&2&3\\3&6&9\end{pmatrix}$.Free
  2. Q10Using Cramer's Rule, solve the system $x+y+z=9$, $2x-y+z=5$, $x+2y-z=4$.Preview
  3. Q11A transition matrix for weather is $P=\begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix}$ (rows/columns ordered Sunny, Rainy). If today's weather…Preview

Sample & Board Papers

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

+Show 33 questions33 questions
  1. Q1If $|A| = 13$ and $|\text{Adj } A| = \begin{vmatrix} 4 & x \\ 5 & 7 \end{vmatrix}$, then the value of $x$ is : (a) $3$ (b) $4$ (c) $2$ (d) $…Preview
  2. Q2Which of the following is not an elementary transformation ? (a) $C_i \rightarrow C_i + 5C_j$ (b) $R_i \leftrightarrow R_j$ (c) $R_i \righta…Preview
  3. Q3The system of linear equations $x + y + z = 2$, $2x + y - z = 3$, $3x + 2y + k = 4$ has unique solution, if $k$ is not equal to : (a) $1$ (b…Preview
  4. Q4The total cost of $11$ pencils and $3$ erasers is $\text{₹ } 64$ and the total cost of $8$ pencils and $3$ erasers is $\text{₹ } 49$. Find t…Preview
  5. Q5If $A = \begin{pmatrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{pmatrix}$ and $B^T = \begin{pmatrix} 1 & -2 & 5 \\ -2 & 4 & 1 \\ 3 & -6…Preview
  6. Q6If $A = \begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}$, then $\rho(A)$ is ______. (a) $2$ (b) $0$ (c) $n$ (d) $1$Preview
  7. Q7If $\rho(A) = \rho(A, B) =$ the number of unknowns, then the system is ______. (a) inconsistent (b) consistent and has infinitely many solut…Preview
  8. Q8Find the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 5 & 7 \end{pmatrix}$.Preview
  9. Q9Consider the matrix of transition probabilities of a product available in the market in two brands A and B. | | A | B | | --- | --- | --- |…Preview
  10. Q10(a) An amount of ₹ 5,000 is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is ₹ 3…Preview
  11. Q11(a) Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previou…Preview
  12. Q12The rank of $m\times n$ matrix whose elements are unity is : (a) $m$ (b) $0$ (c) $n$ (d) $1$Preview
  13. Q13If $|A_{n\times n}|=3$ and $|adj\,A|=243$ then the value of 'n' is : (a) $6$ (b) $4$ (c) $7$ (d) $5$Preview
  14. Q14Find the rank of the matrix $\begin{bmatrix} 0 & -1 & 5 \\ 2 & 4 & -6 \\ 1 & 1 & 5 \end{bmatrix}$Preview
  15. Q15Find k, if the equations $x+2y-3z=-2$, $3x-y-2z=1$, $2x+3y-5z=k$ are consistent.Preview
  16. Q16(a) Solve by Cramer's rule, $x+y+z=4$; $2x-y+3z=1$; $3x+2y-z=1$ OR (b) The population of a certain town is as follows. | Year : X | 1941 | 1…Preview
  17. Q17If $\rho(A)=r$ then which of the following is correct ? (a) A has atleast one minor of order $r$ which does not vanish. (b) A has atleast $(…Preview
  18. Q18If $\frac{a_1}{x}+\frac{b_1}{y}=c_1,\ \frac{a_2}{x}+\frac{b_2}{y}=c_2,\ \Delta_1=\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix},\ \Del…Preview
  19. Q19Show that the equations $3x-2y=6,\ 6x-4y=10$ are inconsistent.Preview
  20. Q20Consider the matrix of transition probabilities of a product available in the market in two brands A and B $\begin{array}{c c} & \begin{arra…Preview
  21. Q21(a) A total of ₹ 8,500 was invested in three interest earning accounts. The interest rates were 2%, 3% and 6%. If the total simple interest…Preview
  22. Q22If $A=\begin{pmatrix}1\\2\\3\end{pmatrix}$, then the rank of $AA^{T}$ is : (a) $2$ (b) $0$ (c) $3$ (d) $1$Preview
  23. Q23If $A$ is a matrix of order $n\times n$, the value of $|adj\,A|$ is : (a) $|A|^{n-1}$ (b) $|A|^{n}$ (c) $|A|^{1+n}$ (d) $|A|$Preview
  24. Q24Find the rank of the given matrix $\begin{bmatrix}3 & 1 & -5 & -1\\1 & -2 & 1 & -5\\1 & 5 & -7 & 2\end{bmatrix}$Preview
  25. Q25The total cost of 11 pencils and 3 erasers is $\text{\rupee}\,64$ and the total cost of 8 pencils and 3 erasers is $\text{\rupee}\,49$. Find…Preview
  26. Q26(a) Solve the following system of equations by rank method : $x+y+z=9,\ 2x+5y+7z=52,\ 2x+y-z=0$ OR (b) Evaluate : $\int \dfrac{3x^{2}-2x+5}{…Preview
  27. Q27(a) Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previou…Preview
  28. Q28If the rank of the matrix $\begin{pmatrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{pmatrix}$ is 2 then $\lambda$ is :…Preview
  29. Q29Rank of a null matrix is : (a) $\infty$ (b) $0$ (c) $1$ (d) $-1$Preview
  30. Q30Find the rank of the matrix $\begin{pmatrix} 1 & 4 & 0 & 5 \\ 9 & 2 & -1 & 4 \end{pmatrix}$Preview
  31. Q31Show that the equations $x - 4y + 7z = 14$, $3x + 8y - 2z = 13$, $7x - 8y + 26z = 5$ are inconsistent.Preview
  32. Q32(a) Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. O…Preview
  33. Q33(a) Solve the equations $x + 2y + z = 7$, $2x - y + 2z = 4$, $x + y - 2z = -1$ by using Cramer's rule. OR (b) Determine an initial basic fea…Preview

More questions

+Show 8 questions8 questions
  1. Example 1Find the rank of $A=\begin{pmatrix}1&2&3\\2&3&4\\3&5&7\end{pmatrix}$.Free
  2. Example 2Find the rank of $B=\begin{pmatrix}1&2&3\\0&1&4\\5&6&0\end{pmatrix}$.Free
  3. Example 3Test the consistency of the system $x+y+z=6$, $x-y+z=2$, $x+2y-z=2$ using the rank method, and solve it if consistent.Free
  4. Example 4Test the consistency of the system $x+y+z=6$, $2x+2y+2z=12$, $x-y+2z=5$ using the rank method.Preview
  5. Example 5Test the consistency of the system $x+y+z=6$, $2x+2y+2z=10$, $x-y+2z=5$ using the rank method.Preview
  6. Example 6Using Cramer's Rule, solve the system $x+y+z=6$, $x+2y+3z=14$, $x+4y+9z=36$.Preview
  7. Example 7The total cost of one unit each of items A, B and C is ₹180. Twice the cost of A plus the cost of B, minus the cost of C, is ₹110. The cost…Preview
  8. Example 8Customers switch between Brand A and Brand B each month according to the transition matrix $P=\begin{pmatrix}0.7&0.3\\0.4&0.6\end{pmatrix}$.…Preview