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Business Mathematics and Statistics · Class 12 Commerce

Ch 2Matrices — Class 12 Business Mathematics and Statistics, concept-first.

This CHSE Odisha Class 11 Commerce Business Mathematics and Statistics chapter introduces matrices — a compact, rectangular way of arranging numbers that businesses use constantly: recording sales of several products across branches, showing costs of raw materials across suppliers, or presenting a firm's assets and lia…

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Key concepts

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Inverse of a Matrix by the Adjoint Method

A square matrix A is non-singular if its determinant A ≠ 0 (and singular if A = 0); only a non-singular matrix has an inverse. The adjoint adj(A) is the transpose of the matrix of cofactors, and A⁻¹ = (1/A)·adj(A).

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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, Order and Notation of a Matrix

This CHSE Odisha Class 11 Commerce Business Mathematics and Statistics chapter introduces matrices — a compact, rectangular way of arranging numbers that businesses use constantly: recording sales of…

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Types of Matrices

Matrices are classified into standard types by their order and the pattern of their elements. Recognising these quickly makes matrix algebra far easier.

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Equality, Addition, Subtraction and Scalar Multiplication of Matrices

Two matrices are equal only if (a) they are of the SAME order, and (b) every corresponding element is equal.

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Multiplication of Matrices

Two matrices (order ) and (order ) can be multiplied, as , only if — i.e. the number of COLUMNS of equals the number of ROWS of . The product then has order .

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Transpose of a Matrix; Symmetric and Skew-Symmetric Matrices

The transpose of a matrix , written (or ), is obtained by interchanging its rows and columns — the first row of becomes the first column of , and so on. If is of order , then is of order .

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Inverse of a Matrix by the Adjoint Method

Finding the inverse of a matrix is one of the most practically useful skills in this CHSE Odisha Business Mathematics and Statistics chapter — it is what makes solving a system of linear equations by…

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Solving a System of Linear Equations by the Matrix Inversion Method

A system of linear equations can be written compactly in matrix form and solved directly using the matrix inverse — a method that generalises cleanly regardless of how many equations are involved.

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Business Applications of Matrices

Matrices are a natural fit for organising the kind of tabular data a business generates every day, and matrix multiplication in particular turns a routine calculation — combining quantities with price…

Exercises

+Show 6 questions6 questions
  1. Q10Find the inverse of $A=\begin{pmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{pmatrix}$ by the adjoint method.Free
  2. Q11Solve the system $3x+2y=8$, $x+4y=6$ by the matrix inversion method.Free
  3. Q12Solve the system $2x-y+z=3$, $-x+2y-z=0$, $x-y+2z=5$ by the matrix inversion method.Preview
  4. Q13A retailer sells Notebooks and Pens at two shops. Shop I sold 50 Notebooks and 80 Pens; Shop II sold 40 Notebooks and 60 Pens. If a Notebook…Preview
  5. Q14If $A$ is a square matrix such that $|A|\neq0$, then $A$ is called: (a) a singular matrix (b) a non-singular matrix (c) a null matrix (d) a…Preview
  6. Q15If $A$ is a square matrix such that $A^T=A$, then $A$ is called: (a) a skew-symmetric matrix (b) an orthogonal matrix (c) a symmetric matrix…Preview

Sample & Board Papers

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

+Show 22 questions22 questions
  1. Q1Express each of the following in one word/term each: (iv) A type of matrix whose number of rows and number of columns are equal.Preview
  2. Q2(f) If $\begin{pmatrix} x-y & z \\ 2x-y & w \end{pmatrix} = \begin{pmatrix} -1 & 4 \\ 0 & 5 \end{pmatrix}$, then the respective values of $x…Preview
  3. Q3(f) If $A = \begin{pmatrix} 2 & 1 \\ 5 & 2 \end{pmatrix}$, $B = \begin{pmatrix} 1 & -3 \\ 2 & 4 \end{pmatrix}$, find $3A + 2B$.Preview
  4. Q4(h) What is meant by order of a matrix?Preview
  5. Q5(l) Find the product of $(2 \quad 3 \quad -1) \times \begin{pmatrix} 1 & 0 & 2 \\ 3 & -2 & 4 \\ 2 & 1 & 0 \end{pmatrix}$.Preview
  6. Q6Using matrix method solve the following system of equations: $2x + 5y = 1$ $3x + 2y = 7$Preview
  7. Q7$\begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ is equal to : (a) $28$ (b) $[20]$ (c) $20$ (d) $[28]$Preview
  8. Q8From the following the one which is an identity matrix, is : (a) $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ (b) $\b…Preview
  9. Q9$(1\ \ 2\ \ 3\ \ 4)$ is an example of : (a) Square matrix (b) Row matrix (c) Column matrix (d) Zero matrixPreview
  10. Q10If $A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}$, then $-3A$ is equal to : (a) $\begin{bmatrix} -6 & -9 \\ 3 & -12 \end{bmatrix}$ (b)…Preview
  11. Q11A matrix $A = [a_{ij}]$ is an unit matrix if : (a) $a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}$ (b…Preview
  12. Q12$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} + \begin{bmatrix} -1 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix} + \begin{bmatrix} 9 & 8 & 7…Preview
  13. Q13What is a singular matrix ?Preview
  14. Q14If $x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, $B = \begin{bmatrix} 8 \\ 1 \\ 4 \end{bmatrix}$, $A^{-1} = \frac{1}{17}\begin{bmatrix} -…Preview
  15. Q15Find the adjoint of the following matrix : $A = \begin{bmatrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{bmatrix}$Preview
  16. Q16Write short notes on : (any two) (a) Multiplication of two matrics (b) Median (c) RangePreview
  17. Q17Express each of the following in one word / term : A matrix that consists of one element onlyPreview
  18. Q18If matrix $A = (2, 3, 4)$ and matrix $B = \begin{pmatrix} 5 \\ 6 \\ 7 \end{pmatrix}$, find $AB$.Preview
  19. Q19From the following matrices, the one which is not a square matrix, is : (a) $[5]$ (b) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ (c) $\b…Preview
  20. Q20If $A = \begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}$, find 2A.Preview
  21. Q21Write the transpose of the matrix $\begin{pmatrix} 7 & 8 & -1 \\ 5 & 2 & 6 \\ 1 & 3 & 4 \end{pmatrix}$.Preview
  22. Q22Solve the following equations by using matrices : $x - y + 2z = 7$ $3x + 4y - 5z = -5$ $2x - y + 3z = 12$Preview

More questions

+Show 9 questions9 questions
  1. Example 1Write down the order of the matrix $B=\begin{pmatrix}5&-2&3\\0&4&-1\end{pmatrix}$, and state the values of the elements $a_{12}$, $a_{23}$ a…Free
  2. Example 2Classify each of the following matrices, giving reasons: $O=\begin{pmatrix}0&0\\0&0\end{pmatrix}$, $D=\begin{pmatrix}3&0\\0&-2\end{pmatrix}$…Free
  3. Example 3If $A=\begin{pmatrix}6&-3\\4&2\end{pmatrix}$ and $B=\begin{pmatrix}-1&5\\3&-4\end{pmatrix}$, find (i) $A+B$ and (ii) $A-B$.Free
  4. Example 4If $A=\begin{pmatrix}2&-3\\0&4\end{pmatrix}$ and $B=\begin{pmatrix}1&2\\-1&3\end{pmatrix}$, find $2A-3B$.Preview
  5. Example 5If $A=\begin{pmatrix}1&2&3\\0&1&4\end{pmatrix}$ (order $2\times3$) and $B=\begin{pmatrix}2&0\\1&3\\0&2\end{pmatrix}$ (order $3\times2$), fin…Preview
  6. Example 6If $A=\begin{pmatrix}2&1\\0&3\end{pmatrix}$ and $B=\begin{pmatrix}1&4\\2&1\end{pmatrix}$, find $AB$ and $BA$, and verify that $AB\neq BA$.Preview
  7. Example 7If $A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$, find the transpose $A^T$, and verify that $(A^T)^T=A$.Preview
  8. Example 8Express the matrix $A=\begin{pmatrix}4&6\\2&8\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.Preview
  9. Example 9Find the inverse of $A=\begin{pmatrix}3&2\\1&4\end{pmatrix}$ by the adjoint method, and verify that $AA^{-1}=I$.Preview