# Matrices Class 12 Maths Chapter 3

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## Certificate

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THIS IS A COMBO OF TWO COURSES

COURSE – 1

In this online course, you will learn concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non- commutativity of multiplication of matrices. Concept of elementary row and column operations and Invertible matrices. This course also has solutions of NCERT Exercise questions. Please note, this course comprises of live session recordings of Ashish Kumar (Agam Sir). Please ignore, if you have heard Agam Sir discussing about live session or doubts session in the course. Live sessions or doubts sessions are no longer offered.

## Course Content

Please note, this course comprises of live session recordings of Ashish Kumar (Agam Sir). Please ignore, if you have heard Agam Sir discussing about live session or doubts session in the course. Live sessions or doubts sessions are no longer offered.

COURSE – 2

This online course is an extended part of Ch03. Matrices Class 12 Maths. The course is based on the assignments by Ashish Kumar (Agam Sir), which have questions from NCERT Exemplar, Board’s Question Bank, R. D. Sharma etc. The PDF of assignments can be downloaded within the course. Please note, the assignments do not have questions based on elementary operations.

## Course Content

The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

• Question 1. Construct a {\displaystyle 2 \times 2} matrix, {\displaystyle A = [{a_{ij}}]} , whose elements are given by: {\displaystyle {a_{ij}} = {e^{2ix}}\sin jx} .
• Question 2. Find the values of x in the following:
{\displaystyle x\, {\begin{bmatrix} {2x}&2\\ 3&x \end{bmatrix}} + 2\, {\begin{bmatrix} 8&{5x}\\ 4&{4x} \end{bmatrix}} = 2\, {\begin{bmatrix} {({x^2} + 8)}&{24}\\ {10}&{6x} \end{bmatrix}} }
• Question 3. Construct a {\displaystyle 3\,\, \times \,3} matrix {\displaystyle {\rm{A }} = \,{[{a_{ij}}]_{\,3\,\, \times \,\,3}}} where {\displaystyle {a_{ij}} = \left\{ \begin{bmatrix}{l} \,\,1 + i + j\,\,\,\,\,\,\,{\rm{if}}\,\,\,i \ge j\\ \,\,\frac{{|i - 2j|}}{2}\,\,\,\,\,\,\,{\rm{if}}\,\,i < j \end{bmatrix} \right.} .
• Question 4. Find the value of x in the following: {\displaystyle {\begin{bmatrix} x&2&{ - 1} \end{bmatrix}} \, {\begin{bmatrix} 1&1&2\\ { - 1}&{ - 4}&1\\ { - 1}&{ - 1}&{ - 2} \end{bmatrix}} \, {\begin{bmatrix} x\\ 2\\ 1 \end{bmatrix}} = [\,0\,]}
• Question 5. Find the matrix X such that, {\displaystyle {\begin{bmatrix} 2&{ - 1}\\ 0&1\\ { - 2}&4 \end{bmatrix}} {\rm{X}} = {\begin{bmatrix} { - 1}&{ - 8}&{ - 10}\\ 3&4&0\\ {10}&{20}&{10} \end{bmatrix}} } .
• Question 6. Find X and Y, if: {\displaystyle 2X + Y + {\begin{bmatrix} { - 2}&1&3\\ 5&{ - 7}&3\\ 4&5&4 \end{bmatrix}} = {\rm{O}}\,\,\,{\rm{;}}\,\,\,X - Y = {\begin{bmatrix} 4&7&0\\ { - 1}&2&{ - 6}\\ { - 2}&8&{ - 5} \end{bmatrix}} }
• Question 7. If {\displaystyle P(x) = {\begin{bmatrix} {\cos x}&{\sin x}\\ { - \sin x}&{\cos x} \end{bmatrix}} } , then show that {\displaystyle P(x).P(y) = P(x + y) = P(y).P(x)} .
• Question 8. Find the value of x in the following: {\displaystyle {\begin{bmatrix} 1&x&1 \end{bmatrix}} \, {\begin{bmatrix} 1&3&2\\ 2&5&1\\ {15}&3&2 \end{bmatrix}} \, {\begin{bmatrix} 1\\ 2\\ x \end{bmatrix}} = {\rm{O}}}
• Question 9. Find the matrix X in the following: {\displaystyle {\begin{bmatrix} 3&2\\ 7&5 \end{bmatrix}} \,X\, {\begin{bmatrix} { - 1}&1\\ { - 2}&1 \end{bmatrix}} = {\begin{bmatrix} 2&{ - 1}\\ 0&4 \end{bmatrix}} }

### Assignment – 2

• Question 1. If {\displaystyle A = {\begin{bmatrix} 4&{x + 2}\\ {2x - 3}&{x + 1} \end{bmatrix}} } is symmetric matrix, then find x.
• Question 2. For what value of x the matrix {\displaystyle {\begin{bmatrix} 0&2&{ - 3}\\ { - 2}&0&{ - 4}\\ 3&4&{x + 5} \end{bmatrix}} } is skew symmetric matrix.
• Question 3. If the matrix {\displaystyle {\begin{bmatrix} 0&a&3\\ 2&b&1\\ c&1&0 \end{bmatrix}} } is a skew- symmetric matrix, find the values of {\displaystyle a,\,\,b\,\,} and {\displaystyle c} .
• Question 4. Express the following matrices as the sum of symmetric and a skew symmetric matrix:
{\displaystyle {\begin{bmatrix} 2&{ - 2}&{ - 4}\\ { - 1}&3&4\\ 1&{ - 2}&{ - 3} \end{bmatrix}} }
• Question 5. If {\displaystyle A = {\begin{bmatrix} 3&{ - 5}\\ { - 4}&2 \end{bmatrix}} } , then find {\displaystyle {A^2} - 5A - 14I} . Hence, obtain {\displaystyle {A^3}} .
• Question 6. What is the number of all possible matrices of order {\displaystyle 2\,\, \times \,\,3} with each entry 0, 1 or 2
• Question 7. If {\displaystyle A = {\begin{bmatrix} {{{\cos }^2}\theta }}&{{\rm{cos \theta sin \theta }}}\\ {{\rm{cos \theta  sin \theta }}}&{{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{ \theta }}} \end{bmatrix}} {\displaystyle {\displaystyle B = {\begin{bmatrix} {{{\cos }^2}\phi }&{{\rm{cos }}\phi {\rm{ sin }}\phi }\\ {{\rm{cos }}\phi {\rm{ sin }}\phi }&{sin}^2\phi } \end{bmatrix}} } then show that AB is a zero matrix, provided {\displaystyle (\theta  - \varphi )} is an odd multiple of {\displaystyle \frac{\pi }{2}} .
• Question 8. If A is a matrix of order {\displaystyle m \times n} and B is a matrix such that {\displaystyle AB'} and {\displaystyle B'A} are both defined, then what is the order of matrix B?
• Question 9. Find x, y, z if {\displaystyle A = {\begin{bmatrix} 0&{2y}&z\\ x&y&{ - z}\\ x&{ - y}&z \end{bmatrix}} } satisfies {\displaystyle A' = {A^{ - 1}}} .

### Assignment – 3

• Question 1. If B be a {\displaystyle 4\,\, \times \,\,5} type matrix, then what is the number of elements in the third column.
• Question 2. If matrix {\displaystyle A = {[{a_{ij}}]_{2 \times 2}},} where {\displaystyle {a_{ij}} = \left\{ \begin{bmatrix}{l} 1 & {\rm{if}}\,i \ne j\\ 0 & {\rm{if}}\,i = j \end{bmatrix} \right.} , then find the value of {\displaystyle {A^2}} .
• Question 3. If {\displaystyle {\rm{A }} = \, {\begin{bmatrix} 4&3\\ 2&5 \end{bmatrix}} } , find {\displaystyle x} and {\displaystyle y} such that {\displaystyle {{\rm{A}}^{\rm{2}}} - x{\rm{A}} + y\,{\rm{I}} = {\rm{0}}} .
• Question 4. Let {\displaystyle A = {\begin{bmatrix} 2&3\\ { - 1}&2 \end{bmatrix}} } . Then show that {\displaystyle {A^2} - 4A + 7I = O} . Using this result find {\displaystyle {A^5}} & {\displaystyle {A^{ - 1}}} .
• Question 5. For the matrix {\displaystyle A = {\begin{bmatrix} 1&1&1\\ 1&2&{ - 3}\\ 2&{ - 1}&3 \end{bmatrix}} } . Show that {\displaystyle {A^3} - 6{A^2} + 5A + 11I = O} . Hence, find {\displaystyle {A^{ - 1}}} .
• Question 6. If A and B are two matrices of the order 3 × m and 3 × n, respectively, and m = n, then what is the order of matrix (5A – 2B)?
• Question 7. If {\displaystyle A = {\begin{bmatrix} {\cos \alpha }&{\sin \alpha }\\ { - \sin \alpha }&{\cos \alpha } \end{bmatrix}} } and {\displaystyle {A^{ - 1}} = A'} , find the value of {\displaystyle \alpha } .

### Assignment – 4

• Question 1. Two shops A and B have tubelights in stock as shown in the table. Shop A places order for 30 Bajaj, 30 Philips, and 20 Surya brand of tubelights, whereas shop B orders 10, 6, 40 numbers of the three varieties. Due to the various factors, they receive only half of the order as supplied by the manufacturers.  The cost of each tubelights of the three types are ₹42, ₹38 and ₹36 respectively.  Represent the following as matrices
1. Initial stock
2. the order
3. the supply
4. final stock
5. cost of individual items (column matrix)
6. total cost of stock in the shops.
• Question 2. A radio manufacturing company produces three models of radios say A, B and C. There is an export order of 500 for model A, 1000 for model B, and 200 for model C. The material and labour (in appropriate units) needed to produce each model is given by the table. Use matrix multiplication to compute the total amount of material and labour needed to fill the entire export order.
• Question 3. There are two families A and B. There are 4 men, 2 women and 1 child in family A and 2 men, 3 women and 2 children in family B.  They recommended daily allowance for calories i.e. Men: 2000, Women: 1500, Children: 1200 and for proteins is Men: 50 gms., Women: 45 , Children: 30 gms. Represent the above information by matrices, using matrix multiplication calculate the total requirements of calories and proteins for each of the families.
• Question 4. There are three families. First family consists of 2 male members, 4 female members and 3 Second family consists of 3 male members, 3 female members and 2 children. Third family consists of 2 male members, 2 female members and 5 children. Male member earns ₹500 per day and spends ₹300 per day. Female member earns ₹400 per day and spends ₹ 250 per day, child member spends ₹40 per day. Find the money each family saves per day using matrices? What is the necessity of saving in the family?
• Question 5. Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold handmade fans, mats and plates from recycled material at a cost of ₹25, ₹100 and ₹50 each. The number of articles sold are given. 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.
• Question 6. An ice-cream stall sells both green tea and mocha ice cream. A small portion of either costs ₹75 and a large portion costs ₹125. During a short period of time, the number of ice creams sold is own in the table. Write down a column matrix, representing the cost of each portion of ice cream. Find total revenue of stall.
• Question 7. The student council is selling flowers for mother’s day. They bought 200 roses for ₹1.67 each, 150 daffodils for ₹1.03 each and 100 orchids for ₹2.59 each.  They sold the roses for ₹3.00 each, the daffodils for ₹2.25 each and the orchids for ₹4.50 each.
1. Organize the data in two matrices, and use matrix multiplication to find the total amount spent of the flowers.
2. Use matrix operations to find the total amount the student council received for the flower sale and how much money the student council made on the project.
• Question 8. A factory produces two models of washing machines, A and B, in three available finishes: N, L and S. Model A is produced in 400 units in Finish N, 200 units in Finish L and 50 units in Finish S. Model B is produced in 300 units in Finish N, 100 units in Finish L and 30 units in Finish S. Finish N takes 25 hours of workshop time to complete and 1 hour of administration. Finish L takes 30 hours of workshop time and 1.2 hours of administration. Finally, Finish S takes 33 hours of workshop time and 1.3 hours of administration. Represent the information in two matrices. Find a matrix that expresses the hours of workshop and administration time needed for each of the models.
Syllabus medium English Hinglish (Hindi + English) 12 Online learning Self-learning from videos Mathematics