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Mathematics · Class 12 Science

Ch 3Matrices — Class 12 Mathematics, concept-first.

The idea of matrices grew directly out of a practical need: a compact, efficient way to solve systems of linear equations. Writing the coefficients of several equations as a single rectangular block turned out to be far more convenient — and far more powerful — than working through elimination by hand, equation by equa…

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Matrix Notation

You run a fruit stall selling apples, bananas, and oranges, and you track sales across Monday, Tuesday, and Wednesday. You could write three separate lists:

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3.1

Introduction

The idea of matrices grew directly out of a practical need: a compact, efficient way to solve systems of linear equations.

3.2

Matrix

A matrix organises information in a rectangular arrangement. To record that Radha has 15 notebooks you could write ; if she also has 6 pens, , with the first number meaning notebooks and the second pe…

3.2.1

Order of a Matrix

The order of a matrix describes its shape: how many rows and how many columns it has.

+Worked Examplesi3 questions
  1. Example 1Consider the following information regarding the number of men and women workers in three factories I, II and III. Men workers / Women worke…Free
  2. Example 2If a matrix has $8$ elements, what are the possible orders it can have?Preview
  3. Example 3Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}\,|\,i - 3j\,|$.Preview
3.3

Types of Matrices

Matrices come in several forms, each defined by its shape or the pattern of its entries. Different operations and properties apply to different kinds, so we examine each type, from the simplest shapes…

3.3.1

Equality of Matrices

12 Q

Two matrices are equal only when they are identical in every respect — a strict condition, not mere similarity of shape. For two matrices and to be equal, two conditions must hold simultaneously.

+Worked Examplesi2 questions
  1. Example 4If $\begin{bmatrix} x+3 & z+4 & 2y-7 \\ -6 & a-1 & 0 \\ b-3 & -21 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ -6 & -3 & 2c+2 \\ 2b+4…Free
  2. Example 5Find the values of $a, b, c,$ and $d$ from the following equation: $\begin{bmatrix} 2a+b & a-2b \\ 5c-d & 4c+3d \end{bmatrix} = \begin{bmatr…Preview
+Exercise 3.1i10 questions
  1. Q1In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write: (i) The or…Free
  2. Q2If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?Free
  3. Q3If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?Free
  4. Q4Construct a $2 \times 2$ matrix, $A = [a_{ij}]$, whose elements are given by: (i) $a_{ij} = \frac{(i+j)^2}{2}$ (ii) $a_{ij} = \frac{i}{j}$ (…Preview
  5. Q5Construct a $3 \times 4$ matrix, whose elements are given by: (i) $a_{ij} = \frac{1}{2} |-3i+j|$ (ii) $a_{ij} = 2i-j$Preview
  6. Q6Find the values of $x, y$ and $z$ from the following equations: (i) $\begin{bmatrix} 4 & 3 \\ x & 5 \end{bmatrix} = \begin{bmatrix} y & z \\…Preview
  7. Q7Find the value of $a, b, c$ and $d$ from the equation: $\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\…Preview
  8. Q8A = $[a_{ij}]_{m \times n}$ is a square matrix, if (A) $m < n$ (B) $m > n$ (C) $m = n$ (D) None of thesePreview
  9. Q9Which of the given values of $x$ and $y$ make the following pair of matrices equal $\begin{bmatrix} 3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}$, $…Preview
  10. Q10Find the value of the following: The number of all possible matrices of order $3 \times 3$ with each entry 0 or 1 is: (A) 27 (B) 18 (C) 81 (…Preview
3.4

Operations on Matrices

Matrices are not just static arrays of numbers; they can be combined and transformed. This section introduces the fundamental operations on matrices — addition, subtraction, scalar multiplication, and…

3.4.1

Addition of Matrices

Consider a manufacturer, Fatima, who owns two factories — one at location A and another at location B.

+Worked Examplesi1 question
  1. Example 6Given $A = \begin{bmatrix} \sqrt{3} & 1 & -1 \\ 2 & 3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2} \…Preview
3.4.2

Multiplication of a Matrix by a Scalar

When a factory doubles its output, every entry in the production matrix is multiplied by 2. This operation — multiplying every element of a matrix by a fixed number — is called scalar multiplication.

+Worked Examplesi1 question
  1. Example 7If $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}$, then find $2…Preview
3.4.3

Properties of Matrix Addition

Matrix addition is the operation of adding two matrices by adding their corresponding entries. For this to be defined, both matrices must be of the same order. If and are both matrices, then

3.4.4

Properties of Scalar Multiplication of a Matrix

Scalar multiplication of a matrix follows algebraic laws that mirror the properties of scalar multiplication in ordinary arithmetic.

+Worked Examplesi4 questions
  1. Example 8If $A = \begin{bmatrix} 8 & 0 \\ 4 & -2 \\ 3 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -2 \\ 4 & 2 \\ -5 & 1 \end{bmatrix}$, then find…Free
  2. Example 9Find $X$ and $Y$, if $X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix}$.Free
  3. Example 10Find the values of $x$ and $y$ from the following equation: $2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 &…Preview
  4. Example 11Two farmers Ramkishan and Gurcharan Singh cultivates only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of…Preview
3.4.5

Multiplication of Matrices

Meera wants 2 pens and 5 story books; Nadeem needs 8 pens and 10 story books. At the first shop a pen costs ₹5 and a story book ₹50, so Meera needs rupees and Nadeem rupees.

+Worked Examplesi4 questions
  1. Example 12Find $AB$, if $A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$.Free
  2. Example 13If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}$, then find $…Free
  3. Example 14If $A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, find $AB$ and $BA$, and show…Preview
  4. Example 15Find $AB$, if $A = \begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}$.Preview
3.4.6

Properties of Multiplication of Matrices

26 Q

Matrix multiplication follows algebraic rules similar to — but not identical with — those for real numbers. The key difference is that it is not commutative: in general, .

+Worked Examplesi4 questions
  1. Example 16If $A = \begin{bmatrix} 1 & 1 & -1 \\ 2 & 0 & 3 \\ 3 & -1 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ 0 & 2 \\ -1 & 4 \end{bmatrix}$ a…Free
  2. Example 17If $A = \begin{bmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{…Free
  3. Example 18If $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}$, then show that $A^3 - 23A - 40I = O$.Preview
  4. Example 19In a legislative assembly election, a political group hired a public relations firm to promote its candidate in three ways: telephone, house…Preview
+Exercise 3.2i22 questions
  1. Q1Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\…Free
  2. Q2Compute the following: (i) $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} + \begin{bmatrix} a & b \\ b & a \end{bmatrix}$ (ii) $\begin{bmatr…Free
  3. Q3Compute the indicated products. (i) $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ (ii) $\beg…Free
  4. Q4If $A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end…Preview
  5. Q5If $A = \begin{bmatrix} \frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3} \end{bmatr…Preview
  6. Q6Simplify $\cos\theta \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta \begin{bmatrix} \sin\the…Preview
  7. Q7Find X and Y, if (i) $X+Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $X-Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$. (ii) $2X…Preview
  8. Q8Find X, if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X+Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$.Preview
  9. Q9Find $x$ and $y$, if $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6…Preview
  10. Q10Solve the equation for $x, y, z$ and $t$, if $2 \begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3 \begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatri…Preview
  11. Q11If $x \begin{bmatrix} 2 \\ 3 \end{bmatrix} + y \begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix}$, find the valu…Preview
  12. Q12Given $3 \begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x+y \\ z+w & 3…Preview
  13. Q13If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$, show that $F(x) F(y) = F(x+y)$.Preview
  14. Q14Show that (i) $\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \neq \begin{bmatrix} 2 & 1 \\ 3 &…Preview
  15. Q15Find $A^2 - 5A + 6I$, if $A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}$.Preview
  16. Q16If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$, prove that $A^3 - 6A^2 + 7A + 2I = 0$.Preview
  17. Q17If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, find $k$ so that $A^2 = kA -…Preview
  18. Q18If $A = \begin{bmatrix} 0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order 2, sh…Preview
  19. Q19A trust fund has ₹ 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bo…Preview
  20. Q20The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are…Preview
  21. Q21Assume $X$, $Y$, $Z$, $W$ and $P$ are matrices of order $2 \times n$, $3 \times k$, $2 \times p$, $n \times 3$ and $p \times k$, respectivel…Preview
  22. Q22Assume $X$, $Y$, $Z$, $W$ and $P$ are matrices of order $2 \times n$, $3 \times k$, $2 \times p$, $n \times 3$ and $p \times k$, respectivel…Preview
3.5

Transpose of a Matrix

A natural operation on a matrix is to flip it — turning rows into columns and columns into rows. This is called transposition, and the result is the transpose of the original.

3.5.1

Properties of Transpose of the Matrices

The transpose of a matrix is a fundamental operation that swaps its rows and columns. If a matrix has entry in the -th row and -th column, then its transpose, denoted (or ), has the entry in the -th r…

+Worked Examplesi2 questions
  1. Example 20If $A = \begin{bmatrix} 3 & 3 & 2 \\ 4 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$, verify that (…Free
  2. Example 21If $A = \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 & -6 \end{bmatrix}$, verify that $(AB)' = B'A'$.Preview
3.6

Symmetric and Skew Symmetric Matrices

13 Q

A square matrix can have special symmetry properties that simplify calculations and reveal structure. The two fundamental types are symmetric and skew symmetric matrices.

+Worked Examplesi1 question
  1. Example 22Express the matrix $B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$ as the sum of a symmetric and a skew symmetr…Preview
+Exercise 3.3i12 questions
  1. Q1Find the transpose of each of the following matrices: (i) $\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$ (ii) $\begin{bmatrix} 1 & -…Free
  2. Q2If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1…Free
  3. Q3If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verif…Free
  4. Q4If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then find $(A + 2B)'$Preview
  5. Q5For the matrices A and B, verify that $(AB)' = B'A'$, where (i) $A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}$, $B = \begin{bmatrix} -1 &…Preview
  6. Q6If (i) $A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, then verify that $A'A = I$ (ii) If $A =…Preview
  7. Q7(i) Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix. (ii) Show that the…Preview
  8. Q8For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (i) $(A + A')$ is a symmetric matrix (ii) $(A - A')$ is a ske…Preview
  9. Q9Find $\frac{1}{2}(A + A')$ and $\frac{1}{2}(A - A')$, when $A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}$Preview
  10. Q10Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) $\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$ (i…Preview
  11. Q11If $A$, $B$ are symmetric matrices of same order, then $AB - BA$ is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Ide…Preview
  12. Q12If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, and $A + A' = I$, then the value of $\alpha$…Preview
3.7

Invertible Matrices

Every non-zero number has a reciprocal, whose product with the number is 1. For matrices there is an analogous idea: the inverse of a square matrix, whose product with the matrix is the identity matri…

+Exercise 3.4i1 question
  1. Q1Matrices $A$ and $B$ will be inverse of each other only if (A) $AB = BA$ (B) $AB = BA = 0$ (C) $AB = 0,\ BA = I$ (D) $AB = BA = I$Preview

Miscellaneous Examples

+Miscellaneous Examplesi3 questions
  1. Example 23If $A = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$, then prove that $A^n = \begin{bmatrix} \cos n\th…Free
  2. Example 24If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB =…Preview
  3. Example 25Let $A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 5 \\ 3…Preview

Miscellaneous Exercise on Chapter 3

+Miscellaneous Exercisei11 questions
  1. Q1If $A$ and $B$ are symmetric matrices, prove that $AB - BA$ is a skew symmetric matrix.Free
  2. Q2Show that the matrix $B'AB$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.Free
  3. Q3Find the values of $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfy the equation $…Free
  4. Q4Find the value of the following: For what values of $x : \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1…Preview
  5. Q5If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, show that $A^2 - 5A + 7I = 0$.Preview
  6. Q6Find $x$, if $\begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x…Preview
  7. Q7A manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below: Market I — $x$: 10,000, $y…Preview
  8. Q8Find the matrix $X$ so that $X\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}…Preview
  9. Q9If $A = \begin{bmatrix} \alpha & \beta \\ \gamma & -\alpha \end{bmatrix}$ is such that $A^2 = I$, then (A) $1 + \alpha^2 + \beta\gamma = 0$…Preview
  10. Q10If the matrix $A$ is both symmetric and skew symmetric, then (A) $A$ is a diagonal matrix (B) $A$ is a zero matrix (C) $A$ is a square matri…Preview
  11. Q11If A is square matrix such that $A^2 = A$, then $(I + A)^3 - 7A$ is equal to (A) A (B) $I - A$ (C) I (D) $3A$Preview

Summary

- A matrix is a rectangular array of numbers arranged in rows and columns, denoted as . - Order of a matrix is ; two matrices are equal only if they have the same order and corresponding elements are…

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 96 questions96 questions
  1. Q1If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?Free
  2. Q2In the matrix $A = \begin{bmatrix} a & 1 & x \\ 2 & \sqrt{3} & x^2 - y \\ 0 & 5 & -\frac{2}{5} \end{bmatrix}$, write: (i) The order of the m…Free
  3. Q3Construct a $2 \times 2$ matrix where (i) $a_{ij} = \dfrac{(i-2j)^2}{2}$ (ii) $a_{ij} = |-2i+3j|$Free
  4. Q4Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = e^{ix}\sin jx$.Preview
  5. Q5Find values of $a$ and $b$ if $A = B$, where $A = \begin{bmatrix} a+4 & 3b \\ 8 & -6 \end{bmatrix}$, $B = \begin{bmatrix} 2a+2 & b^2+2 \\ 8…Preview
  6. Q6If possible, find the sum of the matrices $A$ and $B$, where $A = \begin{bmatrix} \sqrt{3} & 1 \\ 2 & 3 \end{bmatrix}$, and $B = \begin{bmat…Preview
  7. Q7If $X = \begin{bmatrix} 3 & 1 & -1 \\ 5 & -2 & -3 \end{bmatrix}$ and $Y = \begin{bmatrix} 2 & 1 & -1 \\ 7 & 2 & 4 \end{bmatrix}$, find (i) $…Preview
  8. Q8Find non-zero values of $x$ satisfying the matrix equation: $x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 &…Preview
  9. Q9If $A = \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, show that $(A + B)(A - B) \ne…Preview
  10. Q10Find the value of $x$ if $\begin{bmatrix} 1 & x & 1 \end{bmatrix}\begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix}\begin{bm…Preview
  11. Q11Show that $A = \begin{bmatrix} 5 & 3 \\ -1 & -2 \end{bmatrix}$ satisfies the equation $A^2 - 3A - 7I = O$ and hence find $A^{-1}$.Preview
  12. Q12Find the matrix $A$ satisfying the matrix equation: $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bm…Preview
  13. Q13Find $A$, if $\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} A = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}$.Preview
  14. Q14If $A = \begin{bmatrix} 3 & -4 \\ 1 & 1 \\ 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$, then verify…Preview
  15. Q15If possible, find $BA$ and $AB$, where $A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 4 & 1 \\ 2 & 3 \\ 1…Preview
  16. Q16Show by an example that for $A \neq O$, $B \neq O$, $AB = O$.Preview
  17. Q17Given $A = \begin{bmatrix} 2 & 4 & 0 \\ 3 & 9 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ 2 & 8 \\ 1 & 3 \end{bmatrix}$. Is $(AB)'…Preview
  18. Q18Solve for $x$ and $y$: $x\begin{bmatrix} 2 \\ 1 \end{bmatrix} + y\begin{bmatrix} 3 \\ 5 \end{bmatrix} + \begin{bmatrix} -8 \\ -11 \end{bmatr…Preview
  19. Q19If $X$ and $Y$ are $2 \times 2$ matrices, then solve the following matrix equations for $X$ and $Y$: $2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 &…Preview
  20. Q20If $A = \begin{bmatrix} 3 & 5 \end{bmatrix}$, $B = \begin{bmatrix} 7 & 3 \end{bmatrix}$, then find a non-zero matrix $C$ such that $AC = BC$…Preview
  21. Q21Give an example of matrices $A$, $B$ and $C$ such that $AB = AC$, where $A$ is a non-zero matrix, but $B \neq C$.Preview
  22. Q22If $A = \begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ 3 & -4 \end{bmatrix}$ and $C = \begin{bmatrix} 1 & 0 \…Preview
  23. Q23If $P = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ and $Q = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end…Preview
  24. Q24If $\begin{bmatrix} 2 & 1 & 3 \end{bmatrix}\begin{bmatrix} -1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ -1…Preview
  25. Q25If $A = \begin{bmatrix}2 & 1\end{bmatrix}$, $B = \begin{bmatrix}5 & 3 & 4\\ 8 & 7 & 6\end{bmatrix}$ and $C = \begin{bmatrix}-1 & 2 & 1\\ 1 &…Preview
  26. Q26If $A = \begin{bmatrix}1 & 0 & -1\\ 2 & 1 & 3\\ 0 & 1 & 1\end{bmatrix}$, then verify that $A^2 + A = A(A + I)$, where $I$ is the $3 \times 3…Preview
  27. Q27If $A = \begin{bmatrix}0 & -1 & 2\\ 4 & 3 & -4\end{bmatrix}$ and $B = \begin{bmatrix}4 & 0\\ 1 & 3\\ 2 & 6\end{bmatrix}$, then verify that:…Preview
  28. Q28If $A = \begin{bmatrix}1 & 2\\ 4 & 1\\ 5 & 6\end{bmatrix}$ and $B = \begin{bmatrix}1 & 2\\ 6 & 4\\ 7 & 3\end{bmatrix}$, then verify that: (i…Preview
  29. Q29Show that $A'A$ and $AA'$ are both symmetric matrices for any matrix $A$.Preview
  30. Q30Let $A$ and $B$ be square matrices of the order $3 \times 3$. Is $(AB)^2 = A^2 B^2$? Give reasons.Preview
  31. Q31Show that if $A$ and $B$ are square matrices such that $AB = BA$, then $(A + B)^2 = A^2 + 2AB + B^2$.Preview
  32. Q32Let $A = \begin{bmatrix}1 & 2\\ -1 & 3\end{bmatrix}$, $B = \begin{bmatrix}4 & 0\\ 1 & 5\end{bmatrix}$, $C = \begin{bmatrix}2 & 0\\ 1 & -2\en…Preview
  33. Q33If $A = \begin{bmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{bmatrix}$, then show that $A^2 = \begin{bmatrix}\cos 2\theta &…Preview
  34. Q34If $A = \begin{bmatrix}0 & -x\\ x & 0\end{bmatrix}$, $B = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}$ and $x^2 = -1$, then show that $(A + B)…Preview
  35. Q35Verify that $A^2 = I$ when $A = \begin{bmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{bmatrix}$.Preview
  36. Q36Prove by Mathematical Induction that $(A')^n = (A^n)'$, where $n \in \mathbb{N}$ for any square matrix $A$.Preview
  37. Q37If $\begin{bmatrix}xy & 4\\ z+6 & x+y\end{bmatrix} = \begin{bmatrix}8 & w\\ 0 & 6\end{bmatrix}$, then find values of $x$, $y$, $z$ and $w$.Preview
  38. Q38If $A = \begin{bmatrix}1 & 5\\ 7 & 12\end{bmatrix}$ and $B = \begin{bmatrix}9 & 1\\ 7 & 8\end{bmatrix}$, find a matrix $C$ such that $3A + 5…Preview
  39. Q39If $A = \begin{bmatrix}3 & -5\\ -4 & 2\end{bmatrix}$, then find $A^2 - 5A - 14I$. Hence, obtain $A^3$.Preview
  40. Q40Find the values of $a$, $b$, $c$ and $d$, if $3\begin{bmatrix}a & b\\ c & d\end{bmatrix} = \begin{bmatrix}a & 6\\ -1 & 2d\end{bmatrix} + \be…Preview
  41. Q41Find the matrix $A$ such that $\begin{bmatrix}2 & -1\\ 1 & 0\\ -3 & 4\end{bmatrix} A = \begin{bmatrix}-1 & -8 & -10\\ 1 & -2 & -5\\ 9 & 22 &…Preview
  42. Q42If $A = \begin{bmatrix}1 & 2\\ 4 & 1\end{bmatrix}$, find $A^2 + 2A + 7I$.Preview
  43. Q43If $A = \begin{bmatrix}\cos\alpha & \sin\alpha\\ -\sin\alpha & \cos\alpha\end{bmatrix}$, and $A^{-1} = A'$, find value of $\alpha$.Preview
  44. Q44If the matrix $\begin{bmatrix}0 & a & 3\\ 2 & b & -1\\ c & 1 & 0\end{bmatrix}$ is a skew symmetric matrix, find the values of $a$, $b$ and $…Preview
  45. Q45If $P(x) = \begin{bmatrix}\cos x & \sin x\\ -\sin x & \cos x\end{bmatrix}$, then show that $P(x) \cdot P(y) = P(x + y) = P(y) \cdot P(x)$.Preview
  46. Q46If $A$ is square matrix such that $A^2 = A$, show that $(I + A)^3 = 7A + I$.Preview
  47. Q47If $A$, $B$ are square matrices of same order and $B$ is a skew-symmetric matrix, show that $A'BA$ is skew symmetric.Preview
  48. Q48If $AB = BA$ for any two square matrices, prove by mathematical induction that $(AB)^n = A^n B^n$.Preview
  49. Q49Find $x$, $y$, $z$ if $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfies $A' = A^{-1}$.Preview
  50. Q50Express the matrix $\begin{bmatrix} 2 & 3 & 1 \\ 1 & -1 & 2 \\ 4 & 1 & 2 \end{bmatrix}$ as the sum of a symmetric and a skew symmetric matri…Preview
  51. Q51The matrix $P = \begin{bmatrix} 0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0 \end{bmatrix}$ is a (A) square matrix (B) diagonal matrix (C) unit matri…Preview
  52. Q52Total number of possible matrices of order $3 \times 3$ with each entry $2$ or $0$ is (A) $9$ (B) $27$ (C) $81$ (D) $512$Preview
  53. Q53If $\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix} = \begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}$, then the value of $x + y$ is…Preview
  54. Q54If $A = \dfrac{1}{\pi}\begin{bmatrix} \sin^{-1}(\pi x) & \tan^{-1}\left(\dfrac{x}{\pi}\right) \\ \sin^{-1}\left(\dfrac{x}{\pi}\right) & \cot…Preview
  55. Q55If $A$ and $B$ are two matrices of the order $3 \times m$ and $3 \times n$, respectively, and $m = n$, then the order of matrix $(5A - 2B)$…Preview
  56. Q56If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, then $A^2$ is equal to (A) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ (B) $\begin…Preview
  57. Q57If matrix $A = [a_{ij}]_{2 \times 2}$, where $a_{ij} = 1$ if $i \neq j$ and $a_{ij} = 0$ if $i = j$, then $A^2$ is equal to (A) $I$ (B) $A$…Preview
  58. Q58The matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is a (A) identity matrix (B) symmetric matrix (C) skew symmet…Preview
  59. Q59The matrix $\begin{bmatrix} 0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0 \end{bmatrix}$ is a (A) diagonal matrix (B) symmetric matrix (C) skew s…Preview
  60. Q60If $A$ is matrix of order $m \times n$ and $B$ is a matrix such that $AB'$ and $B'A$ are both defined, then order of matrix $B$ is (A) $m \t…Preview
  61. Q61If $A$ and $B$ are matrices of same order, then $(AB' - BA')$ is a (A) skew symmetric matrix (B) null matrix (C) symmetric matrix (D) unit m…Preview
  62. Q62If $A$ is a square matrix such that $A^2 = I$, then $(A-I)^3 + (A+I)^3 - 7A$ is equal to (A) $A$ (B) $I - A$ (C) $I + A$ (D) $3A$Preview
  63. Q63For any two matrices $A$ and $B$, we have (A) $AB = BA$ (B) $AB \neq BA$ (C) $AB = O$ (D) None of the abovePreview
  64. Q64_________ matrix is both symmetric and skew symmetric matrix.Preview
  65. Q65Sum of two skew symmetric matrices is always _________ matrix.Preview
  66. Q66The negative of a matrix is obtained by multiplying it by _________.Preview
  67. Q67The product of any matrix by the scalar _________ is the null matrix.Preview
  68. Q68A matrix which is not a square matrix is called a _________ matrix.Preview
  69. Q69Matrix multiplication is _________ over addition.Preview
  70. Q70If $A$ is a symmetric matrix, then $A^3$ is a _________ matrix.Preview
  71. Q71If $A$ is a skew symmetric matrix, then $A^2$ is a _________.Preview
  72. Q72If $A$ and $B$ are square matrices of the same order, then (i) $(AB)' = $ _________. (ii) $(kA)' = $ _________. ($k$ is any scalar) (iii) $[…Preview
  73. Q73If $A$ is skew symmetric, then $kA$ is a _________. ($k$ is any scalar)Preview
  74. Q74If $A$ and $B$ are symmetric matrices, then (i) $AB - BA$ is a _________. (ii) $BA - 2AB$ is a _________.Preview
  75. Q75If $A$ is symmetric matrix, then $B'AB$ is _________.Preview
  76. Q76If $A$ and $B$ are symmetric matrices of same order, then $AB$ is symmetric if and only if _________.Preview
  77. Q77A matrix denotes a number.Preview
  78. Q78Matrices of any order can be added.Preview
  79. Q79Two matrices are equal if they have same number of rows and same number of columns.Preview
  80. Q80Matrices of different order can not be subtracted.Preview
  81. Q81Matrix addition is associative as well as commutative.Preview
  82. Q82Matrix multiplication is commutative.Preview
  83. Q83A square matrix where every element is unity is called an identity matrix.Preview
  84. Q84If $A$ and $B$ are two square matrices of the same order, then $A + B = B + A$.Preview
  85. Q85If $A$ and $B$ are two matrices of the same order, then $A - B = B - A$.Preview
  86. Q86If matrix $AB = O$, then $A = O$ or $B = O$ or both $A$ and $B$ are null matrices.Preview
  87. Q87Transpose of a column matrix is a column matrix.Preview
  88. Q88If $A$ and $B$ are two square matrices of the same order, then $AB = BA$.Preview
  89. Q89If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.Preview
  90. Q90If $A$ and $B$ are any two matrices of the same order, then $(AB)' = A'B'$.Preview
  91. Q91If $(AB)' = B'A'$, where $A$ and $B$ are not square matrices, then number of rows in $A$ is equal to number of columns in $B$ and number of…Preview
  92. Q92If $A$, $B$ and $C$ are square matrices of same order, then $AB = AC$ always implies that $B = C$.Preview
  93. Q93$AA'$ is always a symmetric matrix for any matrix $A$.Preview
  94. Q94If $A = \begin{bmatrix} 2 & 3 & -1 \\ 1 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}$, then $AB$ an…Preview
  95. Q95If $A$ is skew symmetric matrix, then $A^2$ is a symmetric matrix.Preview
  96. Q96$(AB)^{-1} = A^{-1} \cdot B^{-1}$, where $A$ and $B$ are invertible matrices satisfying commutative property with respect to multiplication.Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 5 questions5 questions
  1. Q1If for three matrices 𝐴 = [𝑎𝑖𝑗]𝑚×4 , B = [𝑏𝑖𝑗]𝑛×3 𝑎𝑛𝑑 C = [𝑐𝑖𝑗]𝑝×𝑞 products 𝐴𝐵 and 𝐴𝐶 both are defined and are square matrices of same order…Preview
  2. Q2The inverse of the matrix $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ is (A) $\begin{bmatrix} 0 & 0 & 3 \\ 0 & 2 & 0…Preview
  3. Q3Assume $X$, $Y$, $Z$, $W$ and $P$ are matrices of order $2\times n$, $3\times k$, $2\times p$, $n\times 3$ and $p\times k$, respectively. Th…Preview
  4. Q4If the matrix $A = \begin{pmatrix} 0 & a & -3 \\ 2 & 0 & -1 \\ b & 1 & 0 \end{pmatrix}$ is skew symmetric, find the values of '$a$' and '$b$…Preview
  5. Q5Given $A = \begin{bmatrix} 2 & -3 \\ -4 & 7 \end{bmatrix}$, compute $A^{-1}$ and show that $2A^{-1} = 9I - A$.Preview