Question

If $2\left( \begin{matrix} 3 & 4 \\\ 5 & x \\\ \end{matrix} \right)+\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right)=\left( \begin{matrix} 7 & 0 \\\ 10 & 5 \\\ \end{matrix} \right)$ find $\left( x-y \right)$.

Answer 1

We know that $k\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)=\left( \begin{matrix} k{{a}_{11}} & k{{a}_{12}} \\\ k{{a}_{21}} & k{{a}_{22}} \\\ \end{matrix} \right)$. Let us assume two matrices $\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$ and $\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$. Let us assume the matrix $\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$ is equal to A. Let us assume the matrix $\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$ is equal to B. Now let us the sum of matrix A and matrix B is equal to matrix C. Now we should find the matrix C.

& \Rightarrow C=A+B \\\ & \Rightarrow C=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)+\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right) \\\ & \Rightarrow C=\left( \begin{matrix} {{a}_{11}}+{{b}_{11}} & {{a}_{12}}+{{b}_{12}} \\\ {{a}_{21}}+{{b}_{21}} & {{a}_{22}}+{{b}_{22}} \\\ \end{matrix} \right) \\\ \end{aligned}$$ We know that if two matrices $$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ and $$\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$$ are said to be equal, if each and every element in the matrix are equal. So, we can say that $$\begin{aligned} & {{a}_{11}}={{b}_{11}} \\\ & {{a}_{12}}={{b}_{12}} \\\ & {{a}_{21}}={{b}_{21}} \\\ & {{a}_{22}}={{b}_{22}} \\\ \end{aligned}$$ By using this concept, we can find the value of $$\left( x-y \right)$$. **Complete step by step answer:** We know that $$k\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)=\left( \begin{matrix} k{{a}_{11}} & k{{a}_{12}} \\\ k{{a}_{21}} & k{{a}_{22}} \\\ \end{matrix} \right)$$. Now we should apply the above condition to the matrix $$2\left( \begin{matrix} 3 & 4 \\\ 5 & x \\\ \end{matrix} \right)$$. $$2\left( \begin{matrix} 3 & 4 \\\ 5 & x \\\ \end{matrix} \right)=\left( \begin{matrix} 6 & 8 \\\ 10 & 2x \\\ \end{matrix} \right)......(1)$$ Now we should know how to add two matrices. Let us assume two matrices $$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ and $$\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$$. Let us assume the matrix $$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ is equal to A. Let us assume the matrix $$\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$$ is equal to B. Now let us the sum of matrix A and matrix B is equal to matrix C. Now we should find the matrix C. $$\begin{aligned} & \Rightarrow C=A+B \\\ & \Rightarrow C=\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)+\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right) \\\ & \Rightarrow C=\left( \begin{matrix} {{a}_{11}}+{{b}_{11}} & {{a}_{12}}+{{b}_{12}} \\\ {{a}_{21}}+{{b}_{21}} & {{a}_{22}}+{{b}_{22}} \\\ \end{matrix} \right) \\\ \end{aligned}$$ In this way, we can find the addition of matrix A and matrix B which is equal to matrix C. Now let us compare $$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ with $$\left( \begin{matrix} 6 & 8 \\\ 10 & 2x \\\ \end{matrix} \right)$$. Now let us compare $$\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$$ with $$\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right)$$. Now we should find the addition of $$\left( \begin{matrix} 6 & 8 \\\ 10 & 2x \\\ \end{matrix} \right)$$ and $$\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right)$$. Let us assume the addition of $$\left( \begin{matrix} 6 & 8 \\\ 10 & 2x \\\ \end{matrix} \right)$$ and $$\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right)$$ is equal to matrix C. Now by using the concept of addition of matrices, we should find the addition of A and B. $$\begin{aligned} & \Rightarrow C=A+B \\\ & \Rightarrow C=\left( \begin{matrix} 6 & 8 \\\ 10 & x \\\ \end{matrix} \right)+\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right) \\\ & \Rightarrow C=\left( \begin{matrix} 6+1 & 8+y \\\ 10+0 & x+1 \\\ \end{matrix} \right) \\\ & \Rightarrow C=\left( \begin{matrix} 7 & 8+y \\\ 10 & x+1 \\\ \end{matrix} \right)........(2) \\\ \end{aligned}$$ From the question, we were given that $$2\left( \begin{matrix} 3 & 4 \\\ 5 & x \\\ \end{matrix} \right)+\left( \begin{matrix} 1 & y \\\ 0 & 1 \\\ \end{matrix} \right)=\left( \begin{matrix} 7 & 0 \\\ 10 & 5 \\\ \end{matrix} \right)......(3)$$ So, from equation (2) and equation (3), we get $$\Rightarrow \left( \begin{matrix} 7 & 8+y \\\ 10 & x+1 \\\ \end{matrix} \right)=\left( \begin{matrix} 7 & 0 \\\ 10 & 5 \\\ \end{matrix} \right)......(4)$$ We know that if two matrices $$\left( \begin{matrix} {{a}_{11}} & {{a}_{12}} \\\ {{a}_{21}} & {{a}_{22}} \\\ \end{matrix} \right)$$ and $$\left( \begin{matrix} {{b}_{11}} & {{b}_{12}} \\\ {{b}_{21}} & {{b}_{22}} \\\ \end{matrix} \right)$$ are said to be equal, if each and every element in the matrix are equal. So, we can say that $$\begin{aligned} & {{a}_{11}}={{b}_{11}} \\\ & {{a}_{12}}={{b}_{12}} \\\ & {{a}_{21}}={{b}_{21}} \\\ & {{a}_{22}}={{b}_{22}} \\\ \end{aligned}$$ So, now by using the concept, we should equate all the elements of matrices in equation (4). So, from equation we can write that $$\begin{aligned} & 8+y=0......(5) \\\ & x+1=5......(6) \\\ \end{aligned}$$ Now from equation (5), we can write that $$\Rightarrow y=-8.....(7)$$ Now from equation (6), we can write that $$\Rightarrow x=4......(8)$$ Now we should find the value of $$\left( x-y \right)$$. Now from equation (7) and equation (8), we get $$\begin{aligned} & \Rightarrow x-y=4-(-8) \\\ & \Rightarrow x-y=12.....(9) \\\ \end{aligned}$$ From equation (9), we get that the value of $$\left( x-y \right)$$ is equal to 12. **Note:** Students should remember that the matrices can be added if both the matrices have the same order. Two matrices of different orders cannot be added by using the concept of addition of matrices. In the same, we can not equate the elements of two matrices if both the matrices have a different order.