Matrices

NCERT Solution

Exercise 1 Part 1

Question 1: In the matrix A = \begin{bmatrix}2 & 5 & 19 & -7\\35 & -2 & 5/2 & 12\\√3 & 1 & -5 & 17\end{bmatrix} write:

(i) The order of the matrix,

(ii) The number of elements,

(iii) Write the elements a13, a21, a33, a24, a23.

Solution:

(i) Since, the given matrix A has three rows and four columns, thus, order of the matrix = 3 x 4

(ii) The given matrix A has 3 x 4 = 12 elements

(iii) a13 = 19, a21 = 35, a33 = –5, a24 = 12, a23 = 5/2.


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

Solution: Since, a matrix is of order m x n has mn elements, thus, possible orders of the having 24 elements will be 1 x 24, 2 x 12, 3 x 8, 4 x 6, 6 x 4, 8 x 3, 12 x 2 and 24 x 1.

Matrix having 13 elements has possible orders are 1 x 13 and 13 x 1

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

Solution: Possible orders of matrix having 18 elements are 1 x 18, 2 x 9, 3 x 6, 6 x 3, 9 x 2, 18 x 1.

Similarly, possible orders of a matrix having 5 elements are 1 x 5 and 5 x 1

Question 4: Construct a `2xx2` matrix, `A=[a_(ij)]` whose elements are given by:

  1. `a_(ij)=((i+j)^2)/2`
  2. `a_(ij)=i/j`
  3. `a_(ij)=((i+2j)^2)/2`

Solution: Let `A_(2xx2)=`\begin{bmatrix}a_(11) & a_(12)\\ a_(21) & a_(22) \end{bmatrix}

(i) `a_(ij)=((i+j)^2)/2`

So, `a_(11)=((1+1)^2)/2=4/2=2`
Similarly, `a_(12)=((2+1)^2)/2=9/2`
`a_(21)=((2+1)^2)/2=9/2`
`a_(22)=((2+2)^2)/2=8`

So, the required matrix can be given as follows:
\begin{bmatrix}2 & 9/2\\ 9/2 & 8\end{bmatrix}

(ii) `a_(ij)=i/j`

Solution: `a_(11)=1/1=1`

`a_(12)=1/2`

`a_(21)=2/1=2`

`a_(22)=2/2=1`

So, the required matrix can be given as follows:

\begin{bmatrix}1 & ½\\2 & 1\end{bmatrix}


(iii) `a_(ij)=((i+2j)^2)/2`

Solution: `a_(11)=((1+2xx1)^2)/2``=((1+2)^2)/2=9/2`

`a_(12)=((1+2xx2)^2)/2``=((1+4)^2)/2=(25)2`

`a_(21)=((2+2xx1)^2)/2``=((2+2)^2)/2=(16)/2=8`

`a_(22)=((2+2xx2)^2)/2``=((2+4)^2)/2=(36)/2=18`

So, the required matrix can be given as follows:

\begin{bmatrix}9/2 & 25/2\\8 & 18\end{bmatrix}



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