Find the best tutors and institutes for Class 12 Tuition
Matrices EXERCISE 3.2 Solutions
- Q1(i): Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find each of the following: (i) $A + B$
- Q1(ii): Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find each of the following: (ii) $A – B$
- Q1(iii): Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find each of the following: (iii) $3A – C$
- Q1(iv): Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find each of the following: (iv) $AB$
- Q1(v): Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$, $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find each of the following: (v) $BA$
- Q2(i): Compute the following: (i) $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} + \begin{bmatrix} a & b \\ b & a \end{bmatrix}$
- Q2(ii): Compute the following: (ii) $\begin{bmatrix} a^2 + b^2 & b^2 + c^2 \\ a^2 + c^2 & a^2 + b^2 \end{bmatrix} + \begin{bmatrix} 2ab & 2bc \\ -2ac & -2ab \end{bmatrix}$
- Q2(iii): Compute the following: (iii) $\begin{bmatrix} -1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5 \end{bmatrix} + \begin{bmatrix} 12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4 \end{bmatrix}$
- Q2(iv): Compute the following: (iv) $\begin{bmatrix} \cos^2 x & \sin^2 x \\ \sin^2 x & \cos^2 x \end{bmatrix} + \begin{bmatrix} \sin^2 x & \cos^2 x \\ \cos^2 x & \sin^2 x \end{bmatrix}$
- Q3(i): Compute the indicated products. (i) $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$
- Q3(ii): Compute the indicated products. (ii) $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} [2 \quad 3 \quad 4]$
- Q3(iii): Compute the indicated products. (iii) $\begin{bmatrix} 1 & -2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}$
- Q3(iv): Compute the indicated products. (iv) $\begin{bmatrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix} \begin{bmatrix} 1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5 \end{bmatrix}$
- Q3(v): Compute the indicated products. (v) $\begin{bmatrix} 2 & 1 \\ 3 & 2 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 1 \end{bmatrix}$
- Q3(vi): Compute the indicated products. (vi) $\begin{bmatrix} 3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 2 & -3 \\ 1 & 0 \\ 3 & 1 \end{bmatrix}$
- Q4: If $A = \begin{bmatrix} 1 & 2 & -3 \\ 5 & 0 & 2 \\ 1 & -1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & -1 & 2 \\ 4 & 2 & 5 \\ 2 & 0 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 4 & 1 & 2 \\ 0 & 3 & 2 \\ 1 & -2 & 3 \end{bmatrix}$, then compute $(A+B)$ and $(B – C)$. Also, verify that $A + (B – C) = (A + B) – C$.
- Q5: If $A = \begin{bmatrix} \frac{2}{3} & 1 & \frac{5}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{7}{3} & 2 & \frac{2}{3} \end{bmatrix}$ and $B = \begin{bmatrix} \frac{2}{5} & \frac{3}{5} & 1 \\ \frac{1}{5} & \frac{2}{5} & \frac{4}{5} \\ \frac{7}{5} & \frac{6}{5} & \frac{2}{5} \end{bmatrix}$, then compute $3A – 5B$.
- Q6: Simplify $\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{bmatrix}$
- Q7(i): Find $X$ and $Y$, if (i) $X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}$ and $X - Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}$
- Q7(ii): Find $X$ and $Y$, if (ii) $2X + 3Y = \begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix}$ and $3X + 2Y = \begin{bmatrix} 2 & -2 \\ -1 & 5 \end{bmatrix}$
- Q8: Find $X$, if $Y = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$ and $2X + Y = \begin{bmatrix} 1 & 0 \\ -3 & 2 \end{bmatrix}$
- Q9: Find $x$ and $y$, if $2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$
- Q10: Solve the equation for $x, y, z$ and $t$, if $2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$
- Q11: If $x\begin{bmatrix} 2 \\ 3 \end{bmatrix} + y\begin{bmatrix} -1 \\ 1 \end{bmatrix} = \begin{bmatrix} 10 \\ 5 \end{bmatrix}$, find the values of $x$ and $y$.
- Q12: Given $3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}$, find the values of $x, y, z$ and $w$.
- Q13: If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$, show that $F(x) F(y) = F(x + y)$.
- Q14(i): Show that (i) $\begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \neq \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & -1 \\ 6 & 7 \end{bmatrix}$
- Q14(ii): Show that (ii) $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \neq \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{bmatrix}$
- Q15: Find $A^2 – 5A + 6I$, if $A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}$
- Q16: If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$, prove that $A^3 – 6A^2 + 7A + 2I = 0$
- Q17: If $A = \begin{bmatrix} 3 & -2 \\ 4 & -2 \end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, find $k$ so that $A^2 = kA – 2I$
- Q18: If $A = \begin{bmatrix} 0 & -\tan\frac{\alpha}{2} \\ \tan\frac{\alpha}{2} & 0 \end{bmatrix}$ and $I$ is the identity matrix of order 2, show that $I + A = (I – A) \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$
- Q19(a): A 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 bond pays 7% interest per year. Using matrix multiplication, determine how to divide ` 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) ` 1800
- Q19(b): A 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 bond pays 7% interest per year. Using matrix multiplication, determine how to divide ` 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (b) ` 2000
- Q20: The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are ` 80, ` 60 and ` 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.
- Q21: Assume $X, Y, Z, W$ and $P$ are matrices of order $2 \times n$, $3 \times k$, $2 \times p$, $n \times 3$ and $p \times k$, respectively. Choose the correct answer. The restriction on $n, k$ and $p$ so that $PY + WY$ will be defined are:
- Q22: Assume $X, Y, Z, W$ and $P$ are matrices of order $2 \times n$, $3 \times k$, $2 \times p$, $n \times 3$ and $p \times k$, respectively. Choose the correct answer. If $n = p$, then the order of the matrix $7X – 5Z$ is:
Other Exercises for CBSE - Class 12 Mathematics Matrices
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Matrices
Top Tutor 
Sector 49, Gurgaon 
12 yrs of Exp 
1 student 
1 hrs of classes completed 
1400 per hour 
Student's Home Focus is on building the fundamentals of both Accounting and economics and making it easy to understand. Have been delivering excellent and above anticipated results. A good score follows with better understanding of the concepts. Learning is fun with the way we do our classes. Must take a demo before making up the mind for starting or not to start the classes.
Sudhanshu Sir is an excellent economics teacher who explains every concept in a very simple and clear way. His teaching style makes even difficult topics easy to understand, and he is always patient while solving doubts. Thanks to his guidance, my interest and confidence in economics have improved a lot.
Top Tutor BRS Nagar, Ludhiana 1 yrs of Exp 27 hrs of classes completed 
Online Classes I was a student in one of the top colleges of Ludhiana. I have proficiency in English . I can teach all subjects to primary wing .
Top Tutor Panduranga Nagar Sahyadri Layout, Bangalore 12 yrs of Exp 8 students 37 hrs of classes completed 800 per hour Online Classes I have 12years of teaching experience in Biology NEET LEVEL. So I can teach according to level of the students and help them to secure good marks in boards as well as neet.
Top Tutor Kalyan East, Thane 8 yrs of Exp 5 students 325 hrs of classes completed 800 per hour Student's Home 
Tutor's home I am profressional in teaching class 12 mathematics since 8 years .Teaching multiple boards and produced excellent results in math.
Super Tutor KK Nagar Sector 10, Chennai 20 yrs of Exp 31 students 2092 hrs of classes completed 700 per hour Online Classes Student's Home Tutor's home I am S.Perumal from Chennai. I am a Assistant Professor of Mathematics. My qualifications are M.Sc., M.Phil., B.Ed. in Mathematics. I am pursuing Ph.D. in mathematics also. Also I am the tutor/trainer of the Urbanpro. Actually, I have very good experience in the field of teaching as more than 23 years. My Qualifications are M.Sc., M.Phil., B.Ed., (Ph.D) in Mathematics. I am allotting 13 regular online classes for a month. I am using zoom app for online classes and using latest technology to write and explain the mathematics problems. My teaching methodology is to make mathematics is easy for students. Initially I give the formulas to students and then I ask them regularly. Also I am solving MCQ. I am teaching mathematics in a simple way. I can make mathematics is easy for students. Always my students got very good marks in exams. I conduct the test and I send the marks to them. Also resolve the students mistakes. After completing every chapters ,i will conduct mathematical quiz for that chapters Revisions: I will revise previous years question papers. After this type of my coaching’s many of my students got 100 marks in their board examination. My teaching type is friendly manner.
It was good, was able to understand the concept I was struggling in and was able to clarify doubts too, barely any issues were network-related, really good for online.
