Question

Find matrix of $A=\left[ \begin{matrix}
   \dfrac{3}{5} & -\dfrac{4}{2} \\
\end{matrix} \right]$ and $B=\left[ \begin{matrix}
   \dfrac{1}{5} & -\dfrac{3}{8} \\
\end{matrix} \right]$ then find $3A-5B+4I$

Answer 1

Hint: We solve this question by assuming that we have the given matrices A and B as 1 x 2 matrices. Then we are required to calculate the value of $3A-5B+4I$ , where I is the identity matrix given by the 1 x 2 matrix $\left[ \begin{matrix}
   1 & 1 \\
\end{matrix} \right].$ Using these, we perform basic matrix operations to obtain the answer.

Complete step-by-step solution:
In order to solve this question, let us first write down the given matrices. The matrices A and B are both 1 x 2 matrices given by,
$\Rightarrow A=\left[ \begin{matrix}
   \dfrac{3}{5} & -\dfrac{4}{2} \\
\end{matrix} \right]$
$\Rightarrow B=\left[ \begin{matrix}
   \dfrac{1}{5} & -\dfrac{3}{8} \\
\end{matrix} \right]$
Next, we need to obtain the value of the expression $3A-5B+4I$ using the basic matrix operations. Here, I is the identity matrix of order 1 x 2 given by,
$\Rightarrow I=\left[ \begin{matrix}
   1 & 1 \\
\end{matrix} \right]$
Substituting these matrices in the expression,
$\Rightarrow 3\times \left[ \begin{matrix}
   \dfrac{3}{5} & -\dfrac{4}{2} \\
\end{matrix} \right]-5\times \left[ \begin{matrix}
   \dfrac{1}{5} & -\dfrac{3}{8} \\
\end{matrix} \right]+4\times \left[ \begin{matrix}
   1 & 1 \\
\end{matrix} \right]$
Multiplying the values outside with each of the element inside the matrix,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5} & -\dfrac{12}{2} \\
\end{matrix} \right]-\left[ \begin{matrix}
   \dfrac{5}{5} & -\dfrac{15}{8} \\
\end{matrix} \right]+\left[ \begin{matrix}
   4 & 4 \\
\end{matrix} \right]$
Simplifying the terms inside by dividing the numbers having a common factor in the numerator and denominator of the first two matrices,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5} & -6 \\
\end{matrix} \right]-\left[ \begin{matrix}
   1 & -\dfrac{15}{8} \\
\end{matrix} \right]+\left[ \begin{matrix}
   4 & 4 \\
\end{matrix} \right]$
Now, we add the corresponding elements of the first two matrices as shown,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5}-1 & -6+\dfrac{15}{8} \\
\end{matrix} \right]+\left[ \begin{matrix}
   4 & 4 \\
\end{matrix} \right]$
The second term is added as $\dfrac{15}{8}$ here since it is negative and it is subtracted so its sign will be positive.
Now, we add the second matrix to the elements of the first matrix as,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5}-1+4 & -6+\dfrac{15}{8} \\
\end{matrix}+4 \right]$
Simplifying the whole numbers by subtracting,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5}+3 & \dfrac{15}{8} \\
\end{matrix}-2 \right]$
Taking the LCM,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5}+\dfrac{3\times 5}{1\times 5} & \dfrac{15}{8} \\
\end{matrix}-\dfrac{2\times 8}{1\times 8} \right]$
Multiplying the terms,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{9}{5}+\dfrac{15}{5} & \dfrac{15}{8} \\
\end{matrix}-\dfrac{16}{8} \right]$
Adding the numerators since the denominators are same,
$\Rightarrow \left[ \begin{matrix}
   \dfrac{24}{5} & -\dfrac{1}{8} \\
\end{matrix} \right]$
Hence, the matrix obtained from the expression $3A-5B+4I$ is $\left[ \begin{matrix}
   \dfrac{24}{5} & -\dfrac{1}{8} \\
\end{matrix} \right].$

Note: We need to know how to perform simple matrix operations in order to solve this question. We need to note an important point here that addition and subtraction of matrices can be done only if they are of the same order. That is why we have considered the identity matrix as a 1 x 2 matrix, else we cannot add or subtract this matrix.