Find the best tutors and institutes for Class 12 Tuition
Q9:
Solve the system of linear equations, using matrix method: $4x – 3y = 3$, $3x – 5y = 7$
Solution :
Step 1: Matrix Representation of the System
The given system of linear equations is:
$4x - 3y = 3$
$3x - 5y = 7$
We represent this system in the matrix form $AX = B$, where:
$A = \begin{bmatrix} 4 & -3 \\ 3 & -5 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, and $B = \begin{bmatrix} 3 \\ 7 \end{bmatrix}$
Step 2: Determinant Calculation and Non-Singularity Check
To determine if the system has a unique solution, we calculate the determinant of matrix $A$ [Per the property of inverse matrices, a unique solution exists if $|A| \neq 0$].
$|A| = \det\begin{bmatrix} 4 & -3 \\ 3 & -5 \end{bmatrix} = (4)(-5) - (-3)(3)$
$|A| = -20 - (-9) = -20 + 9 = -11$
Since $|A| \neq 0$, matrix $A$ is non-singular and invertible. The solution is given by $X = A^{-1}B$.
Step 3: Calculating the Adjoint Matrix $adj(A)$
For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjoint is given by $adj(A) = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
$adj(A) = \begin{bmatrix} -5 & 3 \\ -3 & 4 \end{bmatrix}$
Step 4: Finding the Inverse Matrix $A^{-1}$
Using the formula $A^{-1} = \frac{1}{|A|} adj(A)$:
$A^{-1} = \frac{1}{-11} \begin{bmatrix} -5 & 3 \\ -3 & 4 \end{bmatrix} = \begin{bmatrix} \frac{5}{11} & -\frac{3}{11} \\ \frac{3}{11} & -\frac{4}{11} \end{bmatrix}$
Step 5: Computing the Variable Vector $X$
$X = A^{-1}B = \begin{bmatrix} \frac{5}{11} & -\frac{3}{11} \\ \frac{3}{11} & -\frac{4}{11} \end{bmatrix} \begin{bmatrix} 3 \\ 7 \end{bmatrix}$
Performing matrix multiplication:
$x = (\frac{5}{11})(3) + (-\frac{3}{11})(7) = \frac{15}{11} - \frac{21}{11} = -\frac{6}{11}$
$y = (\frac{3}{11})(3) + (-\frac{4}{11})(7) = \frac{9}{11} - \frac{28}{11} = -\frac{19}{11}$
Final Solution: $x = -\frac{6}{11}, y = -\frac{19}{11}$
More Questions from Class 12 Mathematics Determinants EXERCISE 4.5
- Q1: Examine the consistency of the system of equations: $x + 2y = 2$, $2x + 3y = 3$
- Q2: Examine the consistency of the system of equations: $2x – y = 5$, $x + y = 4$
- Q3: Examine the consistency of the system of equations: $x + 3y = 5$, $2x + 6y = 8$
- Q4: Examine the consistency of the system of equations: $x + y + z = 1$, $2x + 3y + 2z = 2$, $ax + ay + 2az = 4$
- Q5: Examine the consistency of the system of equations: $3x–y – 2z = 2$, $2y – z = –1$, $3x – 5y = 3$
- Q6: Examine the consistency of the system of equations: $5x – y + 4z = 5$, $2x + 3y + 5z = 2$, $5x – 2y + 6z = –1$
- Q7: Solve the system of linear equations, using matrix method: $5x + 2y = 4$, $7x + 3y = 5$
- Q8: Solve the system of linear equations, using matrix method: $2x – y = –2$, $3x + 4y = 3$
- Q10: Solve the system of linear equations, using matrix method: $5x + 2y = 3$, $3x + 2y = 5$
- Q11: Solve the system of linear equations, using matrix method: $2x + y + z = 1$, $x – 2y – z = \frac{3}{2}$, $3y – 5z = 9$
- Q12: Solve the system of linear equations, using matrix method: $x – y + z = 4$, $2x + y – 3z = 0$, $x + y + z = 2$
- Q13: Solve the system of linear equations, using matrix method: $2x + 3y +3 z = 5$, $x – 2y + z = – 4$, $3x – y – 2z = 3$
- Q14: Solve the system of linear equations, using matrix method: $x – y + 2z = 7$, $3x + 4y – 5z = – 5$, $2x – y + 3z = 12$
- Q15: If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$ solve the system of equations $2x – 3y + 5z = 11$, $3x + 2y – 4z = – 5$, $x + y – 2z = – 3$
- Q16: The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is ` 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is ` 70. Find cost of each item per kg by matrix method.
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Determinants
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
5 yrs of
Exp
5 students
851 hrs of classes completed
500
per hour
Online Classes
Total Years of Experience: Over [X years] of experience in teaching across various academic levels, focusing on delivering engaging, student-centered education. 2. Educational Levels Taught: Primary, Secondary, and Senior Secondary levels Undergraduate-level classes (if applicable) Coaching for competitive exams (NEET, JEE, UPSC, etc.) 3. Key Responsibilities: Planning and delivering lessons based on prescribed curriculum. Assessing students’ understanding through regular tests, quizzes, and assignments. Using various teaching methodologies including lectures, group activities, demonstrations, and ICT tools. Providing remedial classes for weak students and enrichment activities for high performers. Maintaining discipline, fostering motivation, and ensuring an inclusive classroom environment. 4. Classroom Management: Successfully managed classrooms of 30–60 students. Promoted active participation and collaborative learning. Ensured timely syllabus completion while maintaining quality of instruction. 5. Communication and Interaction: Maintained regular communication with parents regarding student progress. Participated in parent-teacher meetings, academic planning sessions, and faculty training programs. 6. Technology Integration: Used smart boards, PowerPoint presentations, online quizzes, and digital simulations to enhance learning outcomes. Conducted online classes using platforms like Google Meet, Zoom, and Microsoft Teams during the COVID-19 pandemic. 7. Co-curricular and Administrative Roles: Organized and participated in educational tours, science exhibitions, and debate competitions. Assisted in exam invigilation, results compilation, and record keeping. 8. Achievements and Impact: Improved student performance and pass percentage year-on-year. Received positive feedback from students and parents for teaching effectiveness and approachability. Mentored students who went on to excel in competitive exams and higher education
had a great learning experience under the guidance of my History teacher. Their teaching style is clear, engaging, and well-structured. Complex topics like the Nationalism in India, Partition, and World War themes were explained with such depth and simplicity that it made understanding and remembering easy. They don’t just teach dates and events—they bring history to life with stories, critical thinking, and real-world connections. Regular assessments, personalized feedback, and doubt-clearing sessions really helped me improve my writing and analytical skills. Highly recommended for any Class 12th student aiming for high marks in board exams!
Top Tutor
Tilak Nagar, Delhi
8 yrs of
Exp
4 students
486 hrs of classes completed
800
per hour
Online Classes
I have been teaching dedicatedly for more than eight years and currently working as a Chemistry teacher at Allen Career Institute, Delhi and have been an ex faculty at Fitjee, Chennai. Join online class and fall in love with Chemistry to elevate your score in Board Exams and crack Competitive Exams like JEE Mains & Advanced, NEET etc. and remember that your concepts will be nurtured from the grass root level. 100% fee refund guarantee if you are not satisfied. So book your Chemistry class now and define the Chemistry of life with your own conceptualization.
He is outstanding. I am very happy. Thanks Gaurav and Thanks Urban Pro.Its a very beautiful journey. My son was very week in Chemistry but now after Gaurav take care of my son last couple of months his learning ability continuously grow. Last all exam my ward doing better in Chemistry. I suggest all parents to take a demo class from Gaurav Sir I am definitely sure all of you Choose Gaurav Sir. His professional attitude all together teaching style just Awesome. Rest again I Thanksgiving again Gaurav and urban team.
Top Tutor
Patel Nagar Baljit Nagar, Delhi
7 yrs of
Exp
1 student
24 hrs of classes completed
400
per hour
Online Classes
Student's Home
Concepts will be explained in detail.Marking scheme of Cbse question papers will be discussed. Explanation of case study and technique to crack them. All query will be resolved.
The class is really interactive and informative. The teacher is also really good and patient.
Top Tutor
Jahangirpura, Surat
15 yrs of
Exp
6 students
3746 hrs of classes completed
250
per hour
Online Classes
I have 15 years of teaching Maths to students of clasd 11,12 Science. I have taught NCERT and RD Sharma.
The classes are going well. I am getting a better clarity on the concepts which were already taught. Sir is giving enough time for me to understand better.
Top Tutor
Bejai, Mangalore
15 yrs of
Exp
8 students
209 hrs of classes completed
800
per hour
Online Classes
I am Dr. Shivaprasad Shetty, a Chemistry educator with over 17 years of teaching experience, specializing in Class 12 Chemistry. Teaching has been a long-standing commitment for me, and I focus on helping students develop strong conceptual understanding, analytical thinking, and confidence in the subject. My approach is centered on making Chemistry clear, structured, and easy to apply in examinations. I hold a Doctor of Philosophy (Ph.D.) in Chemistry, which has provided me with advanced academic knowledge and research-based understanding of the subject. My academic training enables me to explain complex chemical concepts with clarity, precision, and depth, ensuring that students build a strong foundation in the subject. Over the years, I have worked as a Chemistry teacher at the college level and have accumulated more than 15 years of teaching experience. During this time, I have consistently delivered strong academic results and received positive feedback from students. I take pride in guiding learners toward better understanding and improved performance in their examinations. My teaching experience has allowed me to work with students of varying academic abilities, helping them overcome challenges in Chemistry and develop confidence in problem-solving. I focus on simplifying difficult topics and ensuring that students grasp the core concepts before moving on to advanced applications. My teaching methodology is based on structured lesson planning with clear objectives for each session. I ensure that every topic is taught in a systematic manner, beginning with fundamentals and gradually progressing to higher-level understanding. I strongly believe that clarity in basics is essential for mastering Chemistry. My classes are interactive and concept-oriented, encouraging students to actively participate and engage with the subject. I also incorporate practical-based explanations and real-life applications wherever possible, which helps students connect theoretical concepts with practical understanding and improves retention. I conduct online classes and use PowerPoint presentations along with whiteboard teaching to make lessons more effective and visually clear. My goal is to create a focused and supportive learning environment where students can strengthen their understanding of Chemistry and achieve academic success with confidence.
Excellent teaching with good clarity . It is interesting and interactive. Easy to understand and classes are having good content with good teaching methodology
