Question

Construct two (2 × 2) matrices P and Q such that the elements of P are given by $a_i_j = i - 2j$ and elements of Q are given by $a_i_j = 2i - j$. Find P + Q ?

Answer 1

Answer:-

A (2 × 2) matrix is the matrix where no of rows = 2 & no.of columns = 2.

Let the elements of matrix P be a₁₁ , a₁₂ , a₂₁ & a₂₂ and Q matrix be a'₁₁ , a'₁₂ , a'₂₁ & a'₂₂

It is given that,

For P matrix, the element:

• aᵢⱼ = i - 2j

So,

• a₁₁ = 1 - 2(1) = 1 - 2 = - 1. [ i = 1 , j = 1 ]

• a₁₂ = 1 - 2(2) = 1 - 4 = - 3. [ i = 1 , j = 2 ]

• a₂₁ = 2 - 2(1) = 2 - 2 = 0. [ i = 2 , j = 1 ]

• a₂₂ = 2 - 2(2) = 2 - 4 = - 2. [ i = 2 , j = 2 ]

Similarly,

For Q matrix, the element:

• aᵢⱼ = 2i - j

So,

• a'₁₁ = 2(1) - 1 = 2 - 1 = 1 [ i = 1 , j = 1 ]

• a'₁₂ = 2(1) - 2 = 2 - 2 = 0 [ i = 1 , j = 2 ]

• a'₂₁ = 2(2) - 1 = 4 - 1 = 3 [ i = 2 , j = 1 ]

• a'₂₂ = 2(2) - 2 = 4 - 2 = 2 [ i = 2 , j = 2 ]

Hence,

$\sf \: P = \begin{bmatrix} \sf \: a_{11} & \sf \: a_{12}\\ \\ \sf \: a_{21}& \sf \: a_{22} \end{bmatrix} = \begin{bmatrix} \sf \: - 1 & \sf \: - 3 \\ \\ \sf \: 0& \sf \: 2 \end{bmatrix}$

And,

$\sf \: Q = \begin{bmatrix} \sf \: a'_{11}& \sf \: a'_{12}\\ \\ \sf \: a'_{21}& \sf \: a'_{22} \end{bmatrix} = \begin{bmatrix} \sf \: 1 & \sf \: 0 \\ \\ \sf \: 3& \sf \: 2 \end{bmatrix}$

Now,

$\implies \sf \: P + Q = \begin{bmatrix} \sf \: - 1 & \sf \: - 3 \\ \\ \sf \: 0& \sf \: - 2 \end{bmatrix} + \begin{bmatrix} \sf \: 1 & \sf \: 0 \\ \\ \sf \: 3& \sf \: 2 \end{bmatrix} \\ \\ \\ \implies \sf \: P + Q =\begin{bmatrix} \sf \: - 1 + 1 & \sf \: - 3 + 0 \\ \\ \sf \:0 + 3& \sf \: - 2 + 2 \end{bmatrix} \\ \\ \\ \implies \red{ \sf \: P + Q = \begin{bmatrix} \sf \: 0 & \sf \: - 3 \\ \\ \sf \:3& \sf \: 0 \end{bmatrix}}$