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Construct a 2 \times 2 matrix, A=\left[a_{i j}\right], whose elements are given by: (i) a_{i j}=\frac{(i+j)^{2}}{2} (ii) a_{i j}=\frac{i}{j} (iii) a_{i j}=\frac{(i+2 j)^{2}}{2}
CBSE | Class 12 | Exercise 3.1 | Maths | Matrices | NCERT | NCERT
Construct a 2 \times 2 matrix, A=\left[a_{i j}\right], whose elements are given by: (i) a_{i j}=\frac{(i+j)^{2}}{2} (ii) a_{i j}=\frac{i}{j} (iii) a_{i j}=\frac{(i+2 j)^{2}}{2}

Solution:

(i) Construct a 2 \times 2 matrix for

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

Elements of 2 \times 2 matrix are: a_{11}, a_{12}, a_{21}, a_{22}

For \mathrm{a}_{11}, \mathrm{i}=1 and \boldsymbol{j}=1
a_{11}=\frac{(1+1)^{2}}{2}=\frac{(2)^{2}}{2}=\frac{4}{2}=2

For \mathbf{a}_{12}, \mathbf{i}=\mathbf{1} and \mathbf{j}=\mathbf{2}

a_{12}=\frac{(1+2)^{2}}{2}=\frac{(3)^{2}}{2}=\frac{9}{2}

For az1, \mathbf{i}=\mathbf{2} and \mathbf{j}=\mathbf{1}
a_{21}=\frac{(2+1)^{2}}{2}=\frac{(3)^{2}}{2}=\frac{9}{2}

For a22, \mathbf{i}=\mathbf{2} and \mathbf{j}=\mathbf{2}

a_{22}=\frac{(2+2)^{2}}{2}=\frac{(4)^{2}}{2}=\frac{16}{2}=8

final matrix is :

\left[\begin{array}{cc}2 & 9 / 2 \\ 9 / 2 & 8\end{array}\right]

(ii) Construct 2 \times 2 matrix for

a_{i j}=\frac{i}{j}

Elements of 2 \times 2 matrix are: a_{11}, a_{12}, a_{21}, a_{22}

For an, \mathbf{i}=\mathbf{1} and \mathbf{j}=\mathbf{1}

a_{11}=\frac{1}{1}=1

For as, \mathbf{i}=\mathbf{1} and \mathbf{j}=\mathbf{2}

a_{12}=\frac{1}{2}

For \mathbf{a}_{21}, \mathbf{i}=\mathbf{2} and \mathbf{j}=\mathbf{1}

a_{21}=\frac{2}{1}=2

For a22, \mathbf{i}=\mathbf{2} and \mathbf{j}=\mathbf{2}

a_{22}=\frac{2}{2}=1

final matrix is

\left[\begin{array}{cc}1 & 1 / 2 \\ 2 & 1\end{array}\right]

(iii) Construct 2 \times 2 matrix for
a_{i j}=\frac{(i+2 j)^{2}}{2}

Elements of 2 \times 2 matrix are: a_{11}, a_{12}, a_{21}, a_{22}

For an, \mathbf{i}=1 and \mathbf{j}=\mathbf{1}
a_{11}=\frac{(1+2)^{2}}{2}=\frac{(3)^{2}}{2}=\frac{9}{2}

For \mathbf{a}_{12}, \mathbf{i}=\mathbf{1} and \mathbf{j}=\mathbf{2} a_{12}=\frac{(1+4)^{2}}{2}=\frac{(5)^{2}}{2}=\frac{25}{2}

For \mathbf{a}_{21}, \mathbf{i}=\mathbf{2} and \mathbf{j}=\mathbf{1}
a_{21}=\frac{(2+2)^{2}}{2}=\frac{(4)^{2}}{2}=\frac{16}{2}=8

For \mathbf{a}_{22}, \mathbf{i}=2 and \mathbf{j}=\mathbf{2}
a_{22}=\frac{(2+4)^{2}}{2}=\frac{(6)^{2}}{2}=\frac{36}{2}=18

final matrix is

\left[\begin{array}{cc}9 / 2 & 25 / 2 \\ 8 & 18\end{array}\right]

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