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