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CBSE Mathematics for Class XII

Dinesh Khattar; Anita Khattar

Chapter 1

Matrices - all with Video Answers

Educators

Section 1

Matrix

01:05

Problem 1

(i) If a matrix has 8 elements, what are the possible orders it can have? What, if it has 5 elements?
(ii) If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?
(iii) How many matrices of order $2 \times 2$ are possible with each entry 0 or $1 ?$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:54

Problem 2

Construct a $2 \times 2$ matrix whose elements $a_{i j}$ are given by
(i) $a_{i j}=\frac{(i-2 j)^{2}}{2}$
(ii) $a_{i j}=\frac{(i+2 j)^{2}}{2}$
(iii) $a_{i j}=\frac{1}{2}|2 i-3 j|$
(iv) $a_{i j}=\frac{1}{2}|-3 i+j|$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
03:54

Problem 3

(i) Construct a $2 \times 2$ matrix whose elements $a_{i j}$ are given by
$$
a_{i j}= \begin{cases}i-j & \text { if } i \geq j \\ i+j & \text { if } i<j\end{cases}
$$
(ii) Construct a $2 \times 2$ matrix whose elements $a_{i j}$ are given by $a_{i j}=\left[\frac{i}{j}\right]$, where $[x]$ stands for greatest integer function.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
06:12

Problem 4

Construct a $3 \times 4$ matrix whose elements $a_{i j}$ are given by
(i) $a_{i j}=i+j-2$
(ii) $a_{i j}=1-3 i-j$
(iii) $a_{i j}=\frac{1}{2}(i+j)^{2}$
(iv) $a_{i j}=\frac{2 i}{j}$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
06:12

Problem 5

(i) Construct a $4 \times 3$ matrix whose elements $a_{i j}$ are given by
$$
a_{i j}= \begin{cases}i^{2} & \text { if } i<j \\ \frac{i}{j} & \text { if } i=j \\ j^{2} & \text { if } i>j\end{cases}
$$
(ii) Construct a $2 \times 2$ matrix whose elements $a_{i j}$ are given by
$$
a_{i j}=\left\{\begin{array}{cl}
i j-j & \text { if } i<j \\
\frac{i}{j} & \text { if } i=j \\
i j-i & \text { if } i>j
\end{array}\right.
$$
(iii) Construct a $2 \times 2$ matrix whose elements are given by
$$
a_{i j}=\left\{\begin{array}{l}
i, \text { when } i \text { is odd } \\
j, & \text { when } i \text { is even }
\end{array}\right.
$$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
06:12

Problem 6

Construct a $3 \times 3$ matrix whose elements $a_{i j}$ are given by
(i) $a_{i j}=i+j$
(ii) $a_{i j}=i-j$
(iii) $a_{i j}=i j$
(iv) $a_{y}=2 i-3 j$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
05:17

Problem 7

If $\left[\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-w\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right]$, find $x, y, z, w .$

AG
Ankit Gupta
Numerade Educator
02:16

Problem 8

Find the values of $x, y, z$ from the following matrix equations:
(i) $\left[\begin{array}{cc}x-3 & 3 x-z \\ x+y+2 & x+y+z\end{array}\right]=\left[\begin{array}{cc}-2 & 0 \\ 5 & 6\end{array}\right]$
(ii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$

Shahab Ullah
Shahab Ullah
Numerade Educator
01:20

Problem 9

If $A=\left[\begin{array}{cc}2 & 3 \\ -1 & 4\end{array}\right]$, find
(i) $2 A$ (ii) $-3 A$.

Abhijith V
Abhijith V
Numerade Educator
00:49

Problem 10

Compute the following sums:
(i) $\left[\begin{array}{cc}2 & -1 \\ 3 & 5\end{array}\right]+\left[\begin{array}{cc}4 & 3 \\ 1 & -2\end{array}\right]$
(ii) $\left[\begin{array}{ccc}0 & 1 & 5 \\ -3 & 2 & 1\end{array}\right]+\left[\begin{array}{ccc}6 & 2 & -3 \\ -1 & 4 & 2\end{array}\right]$
(iii) $\left[\begin{array}{ccc}2 & 3 & 1 \\ 5 & -1 & 2 \\ 0 & 3 & 5\end{array}\right]+\left[\begin{array}{ccc}1 & -2 & 3 \\ -3 & 1 & 5 \\ 6 & 2 & 0\end{array}\right]$
(iv) $\left[\begin{array}{ll}\cos ^{2} x & \sin ^{2} x \\ \sin ^{2} x & \cos ^{2} x\end{array}\right]+\left[\begin{array}{ll}\sin ^{2} x & \cos ^{2} x \\ \cos ^{2} x & \sin ^{2} x\end{array}\right]$
(v) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$
(vi) $\left[\begin{array}{rrr}-1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5\end{array}\right]+\left[\begin{array}{rrr}12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4\end{array}\right]$

AG
Ankit Gupta
Numerade Educator
03:26

Problem 11

If $A=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right], B=\left[\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]$, find each of the following:
(i) $2 B+3 C$
(ii) $-2 A+(B+C)$
(iii) $(2 A-3 B)-C$
(iv) $A+(2 B-C)$
(v) $A+(B+C)$
(vi) $(A+B)+C$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:09

Problem 12

If $A=\left[\begin{array}{ll}2 & 1 \\ 3 & 5 \\ 7 & 9\end{array}\right], B=\left[\begin{array}{ll}3 & 2 \\ 6 & 1 \\ 5 & 4\end{array}\right]$ and $C=\left[\begin{array}{ll}1 & 3 \\ 2 & 0 \\ 3 & 1\end{array}\right]$, find $-5 A+3 B+6 C$.

Shahab Ullah
Shahab Ullah
Numerade Educator
08:13

Problem 13

Express as a single matrix: $4\left[\begin{array}{cc}1 & 3 \\ 1 & -4\end{array}\right]-\frac{1}{2}\left[\begin{array}{ll}8 & 4 \\ 4 & 8\end{array}\right]$

Isaac Chettiath
Isaac Chettiath
Numerade Educator
03:13

Problem 14

(i) If $A=\left[\begin{array}{ccc}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]$, find the matrix $C$ such that $A+B+$
$C$ is a zero matrix.(ii) If $A=\left[\begin{array}{cc}2 & 2 \\ -3 & 1 \\ 4 & 0\end{array}\right], B=\left[\begin{array}{ll}6 & 2 \\ 1 & 3 \\ 0 & 4\end{array}\right]$, find the matrix $C$ such that $A+B+C$ is a null matrix.
(iii) If $A=\left[\begin{array}{ccc}1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1\end{array}\right], B=\left[\begin{array}{ccc}3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3\end{array}\right]$ and $C=\left[\begin{array}{ccc}4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3\end{array}\right]$, then compute $(A+B)$
and $(B-C) .$ Also, verify that $A+(B-C)=(A+B)-C$.

Shahab Ullah
Shahab Ullah
Numerade Educator
08:18

Problem 15

(i) If $A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -2 & 5 & 7\end{array}\right]$ and $2 A-3 B=\left[\begin{array}{ccc}4 & 5 & -9 \\ 1 & 2 & 3\end{array}\right]$, find $B$.
(ii) If $3 X+Y=\left[\begin{array}{ll}3 & 2 \\ 6 & 7 \\ 1 & 9\end{array}\right]$ and $Y=\left[\begin{array}{cc}-6 & 5 \\ 6 & 1 \\ 10 & 0\end{array}\right]$, find $X$.(iii) If $A=\left[\begin{array}{cc}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right]$ and $B=\left[\begin{array}{cc}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$, find the matrix $X$ such that $2 A+3 X=5 B$.
(iv) If $A=\left[\begin{array}{ccc}2 / 3 & 1 & 5 / 3 \\ 1 / 3 & 2 / 3 & 4 / 3 \\ 7 / 3 & 2 & 2 / 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 / 5 & 3 / 5 & 1 \\ 1 / 5 & 2 / 5 & 4 / 5 \\ 7 / 5 & 6 / 5 & 2 / 5\end{array}\right]$, compute $3 A-5 B$.

Urvashi Arora
Urvashi Arora
Numerade Educator
01:03

Problem 16

(i) Find matrices $X$ and $Y$ if $X+Y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$ and $X-Y=\left[\begin{array}{cc}3 & 6 \\ 0 & -1\end{array}\right]$.
(ii) If $A+B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], A-2 B=\left[\begin{array}{cc}-1 & 1 \\ 0 & -1\end{array}\right]$, find $A$ and $B$.
(iii) Find matrices $X$ and $Y$ if $2 X-3 Y=\left[\begin{array}{cc}-7 & 0 \\ 7 & -13\end{array}\right]$ and $3 X+2 Y=\left[\begin{array}{ll}9 & 13 \\ 4 & 13\end{array}\right]$.
(iv) Find matrices $X$ and $Y$ if $3 X+4 Y=\left[\begin{array}{cc}16 & 9 \\ 7 & 24\end{array}\right]$ and $2 X-Y=\left[\begin{array}{ll}7 & 6 \\ 1 & 5\end{array}\right]$.
(v) Find $X$ if $Y=\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]$ and $2 X+Y=\left[\begin{array}{cc}1 & 0 \\ -3 & 2\end{array}\right]$.
(vi) Find $X$ and $Y$, if $2 X+3 Y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$ and $3 X+2 Y=\left[\begin{array}{rr}2 & -2 \\ -1 & 5\end{array}\right]$.
(vii) Find $A+B, A-B$ and $3 A-C$, if $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right]$ and $C=\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$.

NA
Natasha Antonini
Numerade Educator
04:08

Problem 17

If $A=$ diag $[2,-5,9], B=$ diag $[-3,7,14]$ and $C=$ diag $[4,-6,3]$, find
(i) $A+2 B$
(ii) $2 A+B-5 C$

Lucas Finney
Lucas Finney
Numerade Educator
03:28

Problem 18

(i) Find the values of $x$ and $y$ from the equation:
$$
2\left[\begin{array}{cc}
x & 5 \\
7 & y-3
\end{array}\right]+\left[\begin{array}{cc}
3 & -4 \\
1 & 2
\end{array}\right]=\left[\begin{array}{cc}
7 & 6 \\
15 & 14
\end{array}\right]
$$
(ii) Find $x, y, z$ and $w$ if $3\left[\begin{array}{ll}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]$
(iii) Solve the matrix equation:
$$
\left[\begin{array}{ccc}
3 x+1 & 3 & 3 \\
2 & 5 y & 11
\end{array}\right]+2\left[\begin{array}{ccc}
1 & 3 & z-1 \\
2 & 1 & 5
\end{array}\right]=3\left[\begin{array}{ccc}
3 & 3 & 5 \\
2 & 4 & 7
\end{array}\right]
$$
(iv) Find $x, y, z$ and $t$ if
$$
3\left[\begin{array}{ll}
x & y \\
z & t
\end{array}\right]=\left[\begin{array}{cc}
x & 6 \\
-1 & 2 t
\end{array}\right]+\left[\begin{array}{cc}
4 & x+y \\
z+t & 3
\end{array}\right]
$$
(v) Solve the matrix equation for $x$ and $y$ :
$$
\left[\begin{array}{l}
x^{2} \\
y^{2}
\end{array}\right]+2\left[\begin{array}{l}
2 x \\
3 y
\end{array}\right]=3\left[\begin{array}{c}
7 \\
-3
\end{array}\right]
$$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
05:37

Problem 19

If $A=\left[\begin{array}{ccc}3 & 2 & -1 \\ 9 & 5 & 4 \\ 2 & 1 & -6\end{array}\right], B=\left[\begin{array}{ccc}1 & -1 & 4 \\ 2 & 5 & 6 \\ 1 & -3 & 2\end{array}\right]$ and $C=\left[\begin{array}{ccc}4 & -3 & 2 \\ 2 & 1 & 5 \\ 6 & -7 & 8\end{array}\right]$, verify that $(A+B)+$
$C=A+(B+C)$

Shahab Ullah
Shahab Ullah
Numerade Educator
00:32

Problem 20

Find the additive inverse of the following matrices:
(i) $\left[\begin{array}{ccc}1 & 5 & 6 \\ 2 & 7 & 8 \\ 3 & 2 & -1\end{array}\right]$
(ii) $\left[\begin{array}{ccc}-5 & 4 & 0 \\ 7 & -3 & -2\end{array}\right]$
(iii) $\left[\begin{array}{cc}2-3 i & 0 \\ -7 & 1+i\end{array}\right]$

Wesley Hines
Wesley Hines
Numerade Educator
08:18

Problem 21

(i) If $X$ is a matrix and $X+\left[\begin{array}{lll}2 & 3 & 1 \\ 3 & 1 & 2 \\ 1 & 2 & 3\end{array}\right]=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$, find $X$.
(ii) Find the matrix $X$ such that:
$$
\left[\begin{array}{ll}
2 & -3 \\
4 & -2
\end{array}\right]-3 X=\left[\begin{array}{cc}
-3 & 4 \\
5 & -1
\end{array}\right]
$$

Urvashi Arora
Urvashi Arora
Numerade Educator
05:03

Problem 22

Two farmers Ramkishan and Gurcharan Singh cultivate only three varieties of rice namely Basmati, Permal and Naura. The sale (in Rupees) of these varieties of rice by both the farmers in the month of September and October are given by the following matrices $A$ and $B$ :
September Sales (in Rupees)
$$
A=\left[\begin{array}{ccc}
\text { Basmati } & \text { Permal } & \text { Naura } \\
10,000 & 20,000 & 30,000 \\
50,000 & 30,000 & 10,000
\end{array}\right] \text { Ramkishan }
$$
October Sales (in Rupees)
$$
B=\left[\begin{array}{ccc}
\text { Basmati } & \text { Permal } & \text { Naura } \\
5,000 & 10,000 & 6,000 \\
20,000 & 10,000 & 10,000
\end{array}\right] \begin{aligned}
&\text { Ramkishan } \\
&\text { Gurcharan Singh }
\end{aligned}
$$
(i) Find the combined sales in September and October for each farmer in each variety.
(ii) Find the decrease in sales from September to October.
(iii) If both farmers receive $2 \%$ profit on gross sales, compute the profit for each farmer and for each variety sold in October.

Swati Agarwal
Swati Agarwal
Numerade Educator