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Ncert Solutions for Class 12

Matrices

Matrices

Exercise 3.1

1. In the matrix \(A=\begin{bmatrix}2 & 5 & 19 & -7 \\ 35 & -2 &  5\over 2 & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}\), write :

(i) The order of the matrix

(ii) The number of elements

(iii) Write the elements \(a_{13}, a_{21}, a_{33}, a_{24}, a_{23}\).

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

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

4. Construct a 2 × 2 matrix, \(A=[a_{ij}]\), whose elements are given by :

(i) \(a_{ij}={{(i+j)^2} \over 2}\)

(ii) \(a_{ij}={i \over j}\)

(iii) \(a_{ij}={{(i+2j)^2} \over 2}\).

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

(i) \(a_{ij}={1 \over 2}|-3i+j|\)

(ii) \(a_{ij}={2i-j}\)

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

(i) \(\begin{bmatrix}4 & 3 \\ x & 5 \end{bmatrix}\)=\(\begin{bmatrix}y & z \\ 1 & 5 \end{bmatrix}\)

(ii) \(\begin{bmatrix}x+y & 2 \\ 5+z & xy \end{bmatrix}\)=\(\begin{bmatrix}6 & 2 \\ 5 & 8 \end{bmatrix}\)

(iii) \(\begin{bmatrix}x+y+z \\ x+z \\ y+z\end{bmatrix}\)=\(\begin{bmatrix}9\\ 5 \\7 \end{bmatrix}\)

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

\(\begin{bmatrix}a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix}\)=\(\begin{bmatrix}-1 & 5 \\ 0 & 13 \end{bmatrix}\)

8. \(A = {[a_{ij}]}_{m×n}\) is a square matrix, if

(A) m < n
(B) m > n
(C) m = n
(D) None of these

9. Which of the given values of x and y make the following pair of matrices equal

\(\begin{bmatrix}3x+7 & 5 \\ y+1 & 2-3x \end{bmatrix}\), \(\begin{bmatrix}0 & y-2 \\ 8 & 4 \end{bmatrix}\)

(A) \(x={-1\over 3}, y=7\)
(B) Not possible to find
(C) \(y=7, x={-2\over 3}\)
(D) \(x={-1\over 3}, y={-2\over 3}\)

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

Matrices

Exercise 3.2

1. Let \(A=\begin{bmatrix}2 & 4 \\ 3 & 2 \end{bmatrix}\), \(B=\begin{bmatrix}1 & 3 \\ -2 & 5 \end{bmatrix}\), \(A=\begin{bmatrix}-2 & 5 \\ 3 & 4 \end{bmatrix}\) Find each of the following:

(i) A + B
(ii) A – B
(iii) 3A – C
(iv) AB
(v) BA

2. Compute the following:

(i) \(\begin{bmatrix}a & b \\ -b & a \end{bmatrix}\)+\(\begin{bmatrix}a & b \\ b & a \end{bmatrix}\)

(ii) \(\begin{bmatrix}a^2+b^2 & b^2+c^2 \\ a^2+c^2 & a^2+c^2 \end{bmatrix}\)+\(\begin{bmatrix}2ab & 2bc \\ -2ac & -2ab \end{bmatrix}\)

(iii) \(\begin{bmatrix}-1 & 4 & -6 \\ 8 & 5 & 16\\2 & 8 & 5 \end{bmatrix}\)+\(\begin{bmatrix}12 & 7 & 6 \\ 8 & 0 & 5\\3 & 2 & 4 \end{bmatrix}\)

(iv) \(\begin{bmatrix}cos^2x & sin^2x \\ sin^2x & cos^2x\end{bmatrix}\)+\(\begin{bmatrix}sin^2x & cos^2x \\ cos^2x & sin^2x \end{bmatrix}\)

3. Compute the indicated products:

(i) \(\begin{bmatrix}a & b \\ -b & a \end{bmatrix}\)\(\begin{bmatrix}a & -b \\ b & a \end{bmatrix}\)

(ii) \(\begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix}\)\(\begin{bmatrix}2 & 3 & 4 \end{bmatrix}\) 

(iii) \(\begin{bmatrix}1 & -2 \\ 2 & 3 \end{bmatrix}\)\(\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \end{bmatrix}\)

(iv) \(\begin{bmatrix}2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{bmatrix}\)\(\begin{bmatrix}1 & -3 & 5 \\ 0 & 2 & 4\\ 3 & 0 & 5\end{bmatrix}\)

(v) \(\begin{bmatrix}2 & 1 \\ 3 & 2 \\ -1 & 1 \end{bmatrix}\)\(\begin{bmatrix}1 & 0 & 1 \\ -1 & 2 & 1 \end{bmatrix}\)

(vi) \(\begin{bmatrix}3 & -1 & 3 \\ -1 & 0 & 2 \end{bmatrix}\)\(\begin{bmatrix}2 & -3 \\ 1 & 0 \\ 3 & 1 \end{bmatrix}\)

4. 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.

5. If  \(A=\begin{bmatrix}{2\over 3} & 1 & {5\over 3}\\ {1\over 3} & {2\over 3} & {4\over 3} \\ {7\over 3} & 2 & {2\over 3} \end{bmatrix}\) and \(B=\begin{bmatrix}{2\over 5} & {3\over 5} & 1\\ {1\over 5} & {2\over 5} & {4\over 5} \\ {7\over 5} & {6\over 5} & {2\over 5} \end{bmatrix}\), then compute 3A – 5B.

6. Simplify  \(cosθ\begin{bmatrix}cosθ & sinθ \\ -sinθ & cosθ \end{bmatrix}\)+\(sinθ\begin{bmatrix}sinθ & -cosθ \\ cosθ & sinθ \end{bmatrix}\).

7. Find X and Y, if

(i) \(X+Y=\begin{bmatrix}7 & 0 \\ 2 & 5 \end{bmatrix}\) and \(X-Y=\begin{bmatrix}3 & 0 \\ 0 & 3 \end{bmatrix}\)

(ii) \(2X+3Y=\begin{bmatrix}2 & 3 \\ 4 & 0 \end{bmatrix}\) and \(3X+2Y=\begin{bmatrix}2 & -2 \\ -1 & 5 \end{bmatrix}\) 

8. Find X, if  \(Y=\begin{bmatrix}3 & 2 \\ 1 & 4 \end{bmatrix}\) and \(2X+Y=\begin{bmatrix}1 & 0 \\ -3 & 2 \end{bmatrix}\)

9. Find x and y, if  2\(\begin{bmatrix}1 & 3 \\ 0 & x \end{bmatrix}\) + \(\begin{bmatrix}y & 0 \\ 1 & 2 \end{bmatrix}\)=\(\begin{bmatrix}5 & 6 \\ 1 & 8 \end{bmatrix}\)

10. 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\(\begin{bmatrix}3 & 5 \\ 4 & 6 \end{bmatrix}\)

11. 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.

12. Given 3\(\begin{bmatrix}x & y \\ z & w \end{bmatrix}\)=\(\begin{bmatrix}x & 6 \\ -1 & w \end{bmatrix}\)+\(\begin{bmatrix}4 & x+y \\ z+w & 3 \end{bmatrix}\), find the values of x, y, z and w.

13. If  \(F(x)=\begin{bmatrix}cosx & -sinx & 0 \\ sinx & cosx & 0 \\ 0 & 0 & 1 \end{bmatrix}\), show that \(F(x)F(y)=F(x+y)\).

14. Show that 

(i) \(\begin{bmatrix}5 & -1 \\ 6 & 7 \end{bmatrix}\)\(\begin{bmatrix}2 & 1 \\ 3 & 4 \end{bmatrix}\)≠\(\begin{bmatrix}2 & 1 \\ 3 & 4 \end{bmatrix}\)\(\begin{bmatrix}5 & -1 \\ 6 & 7 \end{bmatrix}\)

(ii) \(\begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 0  \\ 1 & 1 & 0\end{bmatrix}\)\(\begin{bmatrix}-1 & 1 & 0 \\ 0 & -1 & 1  \\ 2 & 3 & 4\end{bmatrix}\)≠\(\begin{bmatrix}-1 & 1 & 0 \\ 0 & -1 & 1  \\ 2 & 3 & 4\end{bmatrix}\)\(\begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 0  \\ 1 & 1 & 0\end{bmatrix}\)

15. Find \(A^2-5A+6I\), if \(A=\begin{bmatrix}2 & 0 & 1 \\ 2 & 1 & 3  \\ 1 & -1 & 0\end{bmatrix}\)

16. If \(A=\begin{bmatrix}1 & 0 & 2 \\ 0 & 2 & 1  \\ 2 & 0 & 3\end{bmatrix}\), prove that \(A^3-6A^2+7A+2I=0\)

17. If  \(A=\begin{bmatrix}3 & -2 \\ 4 & -2\end{bmatrix}\) and \(I=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\), find k so that \(A^2=kA-2I\).

18. If  \(A=\begin{bmatrix}0 & -tan{α\over 2} \\ tan{α\over 2} & 0\end{bmatrix}\)  and I is the identity matrix of order 2, show that I+A=(I-A)\(\begin{bmatrix}cosα & -sinα \\ sinα & cosα\end{bmatrix}\).

19. A trust fund has Rs 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 Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of:
(a) Rs 1800
(b) Rs 2000

20. The bookshop of a particular school has 10 dozen chemistry books, 8 dozen 
physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60
and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra.

 

Assume X, Y, Z, W and P are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k,
respectively. Choose the correct answer in Exercises 21 and 22.
21. The restriction 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
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

Matrices

Exercise 3.3

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

(i) \(\begin{bmatrix}5 \\ {1\over 2}\\-1 \end{bmatrix}\)

(ii) \(\begin{bmatrix}1 &-1 \\ 2 & 3 \end{bmatrix}\)

(iii) \(\begin{bmatrix}-1 & 5 & 6 \\ {\sqrt 3} & 5 & 6 \\2 & 3 & -1\end{bmatrix}\)

2. If  \(A=\begin{bmatrix}-1 & 2 & 3 \\ 5 & 7 & 9 \\-2 & 1 & 1\end{bmatrix}\) and \(B=\begin{bmatrix}-4 & 1 & -5 \\ 1 & 2 & 0 \\1 & 3 & 1\end{bmatrix}\), then verify that 

(i) (A+B)’=A’+B’

(ii) (A-B)’=A’-B’

3. If \(A’=\begin{bmatrix}3 & 4 \\ -1 & 2 \\0 & 1\end{bmatrix}\) and \(B=\begin{bmatrix}-1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}\), then verify that

(i) (A+B)’=A’+B’

(ii) (A-B)’=A’-B’

4. If \(A’=\begin{bmatrix}-2 & 3 \\ 1 & 2 \end{bmatrix}\) and \(B=\begin{bmatrix}-1 & 0 \\ 1 & 2 \end{bmatrix}\), then find (A+2B)’.

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

(i) \(A=\begin{bmatrix}1 \\ -4 \\ 3 \end{bmatrix}\), \(B=\begin{bmatrix}-1 & 2 & 1\end{bmatrix}\)

(ii) \(A=\begin{bmatrix}0 \\ 1 \\ 2 \end{bmatrix}\), \(B=\begin{bmatrix}1 & 5 & 7\end{bmatrix}\) 

6. (i) If \(A=\begin{bmatrix}cosα & sinα \\ -sinα & cosα \end{bmatrix}\), then verify that A’A=I.

(ii) If \(A=\begin{bmatrix}sinα & cosα \\ -cosα & sinα \end{bmatrix}\), then verify that A’A=I.

7. (i) Show that the matrix \(A=\begin{bmatrix}1 & -1 & 5 \\ -1 & 2 & 1\\5 & 1 & 3 \end{bmatrix}\) is a symmetric matrix.

(ii) Show that the matrix \(A=\begin{bmatrix}0 & 1 & -1 \\ -1 & 0 & 1\\1 & -1 & 0 \end{bmatrix}\) is a skew symmetric matrix.

8. For the matrix \(A=\begin{bmatrix}1 & 5 \\ 6 & 7 \end{bmatrix}\), verify that 

(i) (A+A’) is a symmetric matrix

(ii) (A-A’) is a skew symmetric matrix.

9. Find  \({1\over 2}(A+A’)\) and \({1\over 2}(A-A’)\), when \(A=\begin{bmatrix}0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}\).

10. Express the following matrices as the sum of a symmetric and a skew symmetric
matrix:

(i) \(\begin{bmatrix}3 & 5 \\ 1 & -1 \end{bmatrix}\)

(ii) \(\begin{bmatrix}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}\)

(iii) \(\begin{bmatrix}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}\)

(iv) \(\begin{bmatrix}1 & 5 \\ -1 & 2 \end{bmatrix}\).

Choose the correct answer in the Exercises 11 and 12.
11. If A, B are symmetric matrices of same order, then AB – BA is a

(A) Skew symmetric matrix
(B) Symmetric matrix
(C) Zero matrix
(D) Identity matrix

12. If  \(A=\begin{bmatrix}cosα & -sinα \\ sinα & cosα \end{bmatrix}\), then A+A’=I, if the value of α is:

(A) \(\pi \over 6\)

(B) \(\pi \over 3\)

(C) \(\pi\)

(D) \({3\pi \over 2}\)

Matrices

Exercise 3.4

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

(A) AB=BA       

(B) AB=BA=0        

(C) AB=0, BA=I 

(D) AB=BA=I.

Matrices

Miscellaneous Exercise

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

2. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

3. Find the values of x, y, z if the matrix \(A=\begin{bmatrix}0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}\) satisfy the equation \(A’A=I\).

4. For what values of x: \(A=\begin{bmatrix}1&2&1\end{bmatrix}\) \(\begin{bmatrix}1&2&0 \\2&0&1\\1&0&2 \end{bmatrix}\)\(\begin{bmatrix}0\\2\\x \end{bmatrix}=O?\)

5. If \(A=\begin{bmatrix}3&1 \\ -1&2\end{bmatrix}\), show that \(A^2-5A+7I=0\).

6. Find x, if \(A=\begin{bmatrix}x&-5&-1\end{bmatrix}\) \(\begin{bmatrix}1&0&2 \\0&2&1\\2&0&3 \end{bmatrix}\)\(\begin{bmatrix}x\\4\\1 \end{bmatrix}=O\).

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

(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 50 paise respectively. Find the gross profit.

8. Find the matrix X so that \(X\begin{bmatrix}1&2&3 \\ 4&5&6 \end{bmatrix}=\)\(\begin{bmatrix}-7&-8&-9 \\ 2&4&6 \end{bmatrix}\).

9. If \(A=\begin{bmatrix}α&β \\ γ &-α\end{bmatrix}\) is such that \(A^2=I\), then

(A) \(1+α^2+βγ=0\)
(B) \(1-α^2+βγ=0\)
(C) \(1-α^2-βγ=0\)
(D) \(1+α^2-βγ=0\).

10. 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

11. If A is square matrix such that \(A^2=A\), then \((I+A)^3–7A\) is equal to
(A) A
(B) I–A
(C) I
(D) 3A