NCERT Solutions for class 12th Mathematics Chapter 3 Matrices

Exercise 3.1

Question 1. In the matrix

Write :

(i) The order of the matrix, (ii) The number of elements,(iii) Write the elements a13, a21, a33, a24, a23.

Question 2. If a matrix has 24 elements, what are the possible orders it can have? What,if it has 13 elements?

Question 3. If a matrix has 18 elements, what are the possible orders it can have ? What,if it has 5 elements .

Question 4. Construct a 2 × 2 matrix, A = [aij] whose elements are given by:

Question 5. Construct a 3 × 4 matrix, whose element are given by:

Question 6. Find the values of x, y, z from the following equations :

Question 7. Find the values of a, b, c and d from the equation:

Question 8. A = [aij]m × n is a square matrix, if

(a) m < n (b) m > n (c) m = n (d) none of these

Sol. (c) For a square matrix: number of rows = number of columns.\ in this case : m = n.

Question 9. Which of the given values of x and y make the following pairs of matrices equal:

Now, from equations (ii) and (iii), we have

5 = y – 2 and y + 1 = 8 Þ y = 8 and y = 7 …(B)

It is clear from equations (A) & (B), we got two different values of x & y both. Only one value of both x and y should satisfy the equations at a time. Hence, no values of x & y satisfies the above equations.

Note: As soon as we got equation (A) in the above question, it was clear that no value of x could satisfy the equation. So it is not necessary to calculate the value of y. We can stop at equation (A) and conclude our result.

Question 10. The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is

(a) 27 (b) 18 (c) 81 (d) 512

Sol. (d) Since the order of matrix is 3 × 3, i.e., it has 9 elements i.e., 9 places to fill. Each element can be selected in 2 ways i.e., either 1 or 0. (given).\9 places can be filled in 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 ways i.e.; 29 ways. i.e.; 512 ways.

Exercise 3.2

Question 1.

Question 2. Compute the following :

Question 3. Compute the indicated products.

Question 7. Find X and Y if

Question 8. Find X

Question 9. Find x and y, if

Question 10. Solve the equation for x, y, z and t, if

Question 11. If

Question 12. Given :

Question 13. F (x) =

Question 14. Show that

Question 15. Find A2 – 5A + 6I,

Question 16. If A =

Question 17. If A =

Question 18. If A =

Question 19. A trust has rs.30,000 that must be invested in two different types of bonds.The first bond pays 5% interest per year and second bond pays 7% interest per year. Using matrix multiplication, determine how to divide rs.30,000 among the two types of bond if the trust fund obtains an annual total interest of (a) rs. 1800 (b) rs. 2000

Question 20. The book-shop of a particular school has 10 dozen Chemistry books, 8 dozen Physics books, 10 dozen Economics books. Their selling price are rs. 80,rs. 60 and rs. 40 each respectively. Find the total amount the book-shop willreceive from selling all the books using matrix algebra.

Question 21. The restrictions on n, k and p so that PY + WY will be defined are (a) k = 3, p = n (b) k is arbitrary, p = 2(c) p is arbitrary, k = 3 (d) k = 2, p = 3

Question 22. If n = p, then the order of the matrix 7X – 5Z is :

(a) p × 2 (b) 2 × n (c) n × 3 (d) p × n

Exercise 3.3

Question 1. Find the transpose of each of the following matrices :

Question 2. If A =

Question 3. If A’=

Question 4. If A’=

Question 5. For the matrices A and B, verify that (AB)’ = B’A’, where

Question 6.

Question 7.

Question 8.

Question 9.

Question 10. Express the following matrices as the sum of a symmetric and a skewsymmetric matrix

Question 11. Choose the correct answer in the following questions :

If A, B are symmetric matrices of same order then AB – BA is

(a) Skew – symmetric matrix (b) Symmetric matrix (c) Zero matrix (d) Identity matrix

Question 12.

Exercise 3.4

Using Elementary transformation, find the inverse of each of matrices, if it exists in questions 1 to 17

Note: Elementary column operations can also be used to find the inverse of a matrix in the same way as above.

Note:

While calculating inverse of a matrix by elementary row operation, if we get every element of any row as zero in any step, than the inverse of the given matrix does not exist.Same is the case for elementary column operation (any column as zero)

Question 18. Matrices A and B will be inverse of each other only if

(a) AB = BA (b) AB = BA = 0 (c) AB = 0, BA = 1 (d) AB = BA = I

Miscellaneous Exercise

Question 1.

Hint : Use principle of Mathematical Induction

Question 2.

Question 3.

Question 4. If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.

Question 5. Show that the matrix B¢ AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Question 6. Find the values of x, y, z if the matrix A =

Satisfy the Equation A’A= I

Question 7. For what values of x :

Question 8. if A =

Question 9.

Question 10. A manufacturer produces three products x, y, z which he sells in two markets.Annual sales are indicated below :

MarketProducts
I10,0002,00018,000
II6,00020,0008,000

(a) If unit sale prices of x, y and z are rs.2.50, rs.1.50 and rs.1.00, respectively,find the total revenue in each market with the help of matrix algebra.

(b) If the unit costs of the above three commodities are rs.2.00, rs.1.00 and 50paise respectively. Find the gross profit.

Question 11. Find the matrix X so that

Question 12. If A and B are square matrices of the same order such that AB = BA, thenprove by induction that ABn = BnA. Further, prove that (AB)n= AnBn for all n E N.

Question 13.

Question 14. If the matrix A is both symmetric and skew symmetric, then

(a) A is a diagonal matrix (b) A is a zero matrix (c) A is a square matrix (d) None of these

Question 15. If A is square matrix such that A2 = A, then (I + A)3 – 7 A is equal to (a) A (b) I – A (c) I (d) 3A

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