Matrices(Ex. 3.2)

Exercise 3.2

1. 

    

Find each of following:

(i) A+B

Solun:- According to the addition law of matrix:-

Because Order of A = Order of B

So, the addition of matrices is possible.

(ii) A-B

Solun:- Because Order of A = Order of B

So, the difference of matrices is possible.

(iii) 3A-C

Solun:- Calculate 3A:-

Because Order of 3A = Order of C

So, the difference of matrices is possible.

(iv) AB

Solun:- No. of columns of matrix A =2

No. of rows of matrix B = 2

No. of columns of matrix A = No. of rows of matrix B

Then the multiplication of matrices is possible.

(v) BA

Solun:-  No. of columns of matrix B =2

No. of rows of matrix A = 2

No. of columns of matrix B = No. of rows of matrix A

Then the multiplication of matrices is possible.

2. Compute the following:

3. Compute the indicated products:-

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

Solun:- No. of columns of first matrix = No. of rows of the second matrix

then matrix multiplication is possible.

 

then compute (A+B) and (B-C). Also, verify that A+(B-C) = (A+B)-C.

Solun:- By definition of the addition of matrices:- Order of matrix A=Order of matrix B

then the addition of matrices is possible.

Taking L.H.S:-

Taking R.H.S:-

L.H.S = R.H.S (Hence Proved……....)

5. Compute 3A-5B.

 

Solun:- Calculate 3A:-

Calculate 5B:-

Order of matrix 3A = Order of matrix 5B

then calculate 3A-5B

7. Find X and Y, if

Order of both the matrices is same then compare corresponding elements.

⇒ 2+y=5

⇒ y=3

⇒ 2x+2=8

⇒ x+1=4

⇒ x=3

⇒ x=3 and y=4

10. Solve these equations for x,y,z and t, if:

Order of both the matrices is the same then compare corresponding elements.

⇒ x=3, y=6, z=9 and t=6

Find the values of x and y.

Order of both the matrices is the same then compare corresponding elements.

⇒ 2x-y=10…………(1)

⇒ 3x+y=5   (By elimination method)

⇒ 5x=15

⇒ x=3

Put the value of x in Eq. 1:-

⇒ 6-y=10

⇒ y=-4

⇒ x = 3 and y = -4

Find the values of x,y,z and w.

Order of both the matrices is the same then compare corresponding elements.

⇒ 3x=x+4

⇒ 2x=4

⇒ x=2

⇒ 3y=6+x+y

⇒ 2y=8

⇒ y=4

⇒ 3w=2w+3

⇒ w=3

⇒ 3z=-1+z+w

⇒ 2z= 2

⇒ z=1

⇒ x=2, y=4,w=3 and z=1

We know cos(x+y)=cosxcosy-sinxsiny

⇒ sin(x+y)=sinxcosy+cosxsiny

Put these values in the matrix:-

Calculate F(x+y):-

So, F(x)F(y)=F(x+y) (Hence Proved…....)

14. Show that

Solun:- Taking L.H.S:-

Taking R.H.S:-

Thus R.H.S doesn’t equal to L.H.S.

Taking R.H.S:-

Thus RHS doesn’t equal to LHS.

Solun:- Given Eq. is A2-5A+6I   then Calculate A2

Put all values in given Eq. :-

Solun:- Given equation is:- A3-6A2+7A+2I=0

(I is the identity matrix of order 3 because for addition same order matrix is required)

Calculate A3:-

Put all the values in a given equation:-

L.H.S=R.H.S (Hence Proved…....)

Solun:- Given eq. is:- A2=kA-2I  Calculate A2:-

Put values in given equation:-

Solun:- Taking L.H.S:- I+A

So, L.H.S = R.H.S (Hence Proved…....)

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

(A) Rs 1800

(B) Rs 2000

Solun:- Given funds must be invested in two types of bonds.

Let the first bond is=x

Second bond is=30000-x

Investment is:-

Rate is:-

(i) Obtain total annual interest is:-

So the first investment is Rs 15000 and the second investment is 30000-15000=Rs 15000.

(ii) Obtain total annual interest is:-

So the first investment is Rs 5000 and the second investment is 30000-5000=Rs 25000.

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

Solun:- Chemistry books=10 dozen =10x12=120

Physics books=8 dozen =8x12=96

Economics books=10 dozen =10x12=120

Total amount the bookshop is:-

=Rs 20160

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

Solun:- Given PY+WY=Y(P+W)

For the addition of P and W:-P and W should have the same order.

That is:- pxk=nx3

So n=p and k=3  (Comparing both sides)

For multiplication of Y(P+W)

No. of columns of Y=No. of rows of P+W

⇒ k=n=p=3

So the answer is A.

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

(A) px2

(B) 2xn

(C) nx3

(D) pxn

Solun:- Order of X is=2xn

And the order of Z is=2xp=2xn  (Given n=p)

Order of 7X-5Z =2xn. (Because result matrix has same order that has X and Z)

The Answer is……B

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