Matrices

Previous Years' CBSE Board Questions

3.2 Matrix

(1 mark)

Construct a <math>2 \times 2</math> matrix <math>A = [a_{ij}]</math>, whose elements are given by <math>a_{ij} = |(i)^2 - j|</math>. (2020)

2 Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3. (Al 2016)

1. Write the element <math>a_{23}</math> of a <math>3 \times 3</math> matrix <math>A = [a_{ij}]</math> whose elements <math>a_{ij}</math> are given by <math>a_{ij} = \frac{|i-j|}{2}</math>. (Delhi 2015)
2. The elements <math>a_{ij}</math> of a 3 <math>\times</math> 3 matrix are given by <math>a_{ij} = \frac{1}{2} |-3i+j|</math>. Write the value of element <math>a_{32}</math>. (AI 2014C)

3.3 Types of Matrices

(1 mark)

1. If <math>\begin{bmatrix} x-y & z \\ 2x-y & w \end{bmatrix} = \begin{bmatrix} -1 & 4 \\ 0 & 5 \end{bmatrix}</math>, find the value of <math>x+y</math>. (Al 2014) (Ap)
2. If <math>\begin{pmatrix} a+4 & 3b \\ 8 & -6 \end{pmatrix} = \begin{pmatrix} 2a+2 & b+2 \\ 8 & a-8b \end{pmatrix}</math>, write the value of a - 2b. (Foreign 2014) Ap
3. If <math>\begin{bmatrix} x \cdot y & 4 \\ z+6 & x+y \end{bmatrix} = \begin{bmatrix} 8 & w \\ 0 & 6 \end{bmatrix}</math>, write the value of <math>(x+y+z)</math>. (Delhi 2014C)

3.4 Operations on Matrices

MCQ

1. If <math>A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}</math>, <math>B = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}</math> and <math>A = B^2</math>, then x equals
2. ±1 (b) -1 (c) 1 (d) 2 (2023)

1. If <math>\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 3 \\ 2 \end{bmatrix}</math>, then the value of <math>(2x + y - z)</math> is
2. 1 (b) 2 (c) 3 (d) 5 (2023)

1. If <math>x \begin{vmatrix} 1 \\ 2 \end{vmatrix} + y</math> <math>\begin{bmatrix} 2 \\ 5 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}</math>, then
2. <math>x = 1, y = 2</math> (b) <math>x = 2, y = 1</math>
3. <math>x = 1, y = -1</math> (d) <math>x = 3, y = 2</math>

1. If A is a square matrix and A² = A, then (I + A)² - 3A is equal to
2. I (b) A (c) 2A (d) 31

(2023)

12. If <math>A = \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}</math>, then (A - 2I)(A - 3I) is equal to

(a) A (b) I (c) 5I

(Term I, 2021-22)

If order of matrix A is <math>2 \times 3</math>, of matrix B is <math>3 \times 2</math>, and

of matrix C is 3 × 3, then which one of the following is

not defined?

1. C(A + B') (b) C(A + B')'
2. BAC (d) CB + A' (Term I, 2021-22)

14. If <math>A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -1 & 1 \\ 1 & -1 & 1 \end{bmatrix}</math>, then <math>A^5 - A^4 - A^3 + A^2</math> is equal to

1. 2A (b) 3A (c) 4A (Term I, 2021-22)

If A is a square matrix such that A² = A, then (I - A)³ + A

is equal to

1. I
2. I-A (d) I+A (2020)

AB + XY equals (a) [28] 16. If <math>A = \begin{bmatrix} 2 & -3 & 4 \end{bmatrix}</math>, <math>B = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix}</math>, <math>X = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}</math> and <math>Y = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}</math>, then (b) [24] (c) 28 (d) 24 (2020)

VSA (1 mark)

17. If <math>A = \begin{bmatrix} 1 & 0 & 4 \end{bmatrix}</math> and <math>B = \begin{bmatrix} 5 \\ 4 \end{bmatrix}</math>, find AB. (2021) Ev

Find the order of the matrix A such that

<math display="block">\begin{vmatrix} 2 & -1 \\ 1 & 0 \\ -3 & 4 \end{vmatrix} = \begin{vmatrix} -1 & -8 \\ 1 & -2 \\ 9 & 22 \end{vmatrix}.</math> (2021) R

1. If <math>B = \begin{bmatrix} 1 & -5 \\ 0 & -3 \end{bmatrix}</math> and <math>A + 2B = \begin{bmatrix} 0 & 4 \\ -7 & 5 \end{bmatrix}</math>, find the matrix (2021)
2. If <math>A + B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}</math> and <math>A - 2B = \begin{bmatrix} -1 & 1 \\ 0 & -1 \end{bmatrix}</math>, then (2020)
3. If A is a square matrix such that A2 = I, then find the simplified value of <math>(A - I)^3 + (A + I)^3 - 7A</math>. (NCERT Exemplar, Delhi 2016) (An

0 -1][1

1. If [2 1 3] -1 1 0 0 = A, then write the 0 1 1 ][-1]

order of matrix A. (Foreign 2016) (Ap)

Solve the following matrix equation for x: <math>\begin{bmatrix} x & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ -2 & 0 \end{bmatrix} = 0</math> (Delhi 2014)

1. If <math>2\begin{bmatrix} 3 & 4 \\ 5 & x \end{bmatrix} + \begin{bmatrix} 1 & y \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0 \\ 10 & 5 \end{bmatrix}</math>, find (x - y). (Delhi 2014)

Find the funds collected by each school separately

by selling the above articles. Also, find the total funds

collected for the purpose. Write one value generated by the above situation.

(Delhi 2015)

1. If A is a square matrix such that A2 = A, then write the value of <math>7A - (I + A)^3</math>, where I is an identity matrix. (Al 2014) Ev
2. If <math>(2x \ 4) \begin{pmatrix} x \\ -8 \end{pmatrix} = 0</math>, find the positive value of x. (AI 2014C)

SAI (2 mark)

To promote the making of toilets for women, an

organisation tried to generate awareness through

(i) house calls (ii) letters and (iii) announcements.

The cost for each mode per attempt is given

below:

1. ₹50 (ii) ₹ 20 (iii) ₹ 40 The number of attempts made in three villages X, Y and Z are given below:
2. (iii) (iiii)

X 400 300 100

300 250 75

500 400 7 150

1. If <math>A = \begin{bmatrix} -3 & 2 \\ 1 & -1 \end{bmatrix}</math> and <math>I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math>, find scalar k so that <math>A^2 + I = kA</math> (2020)

1. For what value of x is <math>\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ 0 & 1 \end{bmatrix} = 0</math>? (2020) Ev

Find a matrix A such that 2A - 3B + 5C = O, where <math display="block">B = \begin{bmatrix} -2 & 2 & 0 \\ 3 & 1 & 4 \end{bmatrix} \text{ and } C = \begin{bmatrix} 2 & 0 & -2 \\ 7 & 1 & 6 \end{bmatrix}. \text{ (Delhi 2019)} </math>

SAII (3 mark)

1. If <math>A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}</math>, then show that <math>A^3 - 23A - 40I = 0</math>. (2023)
2. In a parliament election, a political party hired a public relations firm to promote its candidates in three ways-telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as 140 Telephone

LAI (4 marks)

1. Let <math>A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix} B = \begin{pmatrix} 5 & 2 \\ 7 & 4 \end{pmatrix} C = \begin{pmatrix} 2 & 5 \\ 3 & 8 \end{pmatrix}</math> find a matrix D such that <math>CD - AB = O</math>. (Delhi 2017) 📶

1 0 A= 1 -2

1. Find matrix A such that The number of contacts of each type made in two cities X and Y is given in matrix B as Telephone Housecall Letters <math>B = \begin{bmatrix} 1000 & 500 \\ 3000 & 1000 \end{bmatrix}</math> 10000 City Y 5000 City X Find the total amount spent by the party in the two cities. What should one consider before casting his/her vote-party's promotional activity or their

social activities? (Foreign 2015)

1. If <math>A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}</math>, find <math>A^2 - 5A + 4I</math> and hence find a matrix X such that <math>A^2 - 5A + 4I + X = O</math> (Delhi 2015) An

1. Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of ₹ 25, ₹ 100 and ₹ 50 each. The number of articles sold are given below.

Article/School A В C

Hand-fans 40 25 35

Mats 50 40 50

Plates 20 30 40

Find the total cost incurred by the organisation for

the three villages separately, using matrices. Write

one value generated by the organisation in the

society. (Al 2015) (Ev

36. If <math>A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}</math> and <math>B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}</math> and <math>(A + B)^2 = A^2 + B^2</math>,

then find the values of a and b. (Foreign 2015)

A = 200 House call 150 Letters

38. If <math>\begin{bmatrix} 2x & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & x \\ -3 & 0 & 3 \end{bmatrix} = 0</math>, find x. (Delhi 2015C)

A trust fund, ₹ 35,000 is to be invested in two

different types of bonds. The first bond pays

8% interest per annum which will be given to