Question

$\left[ {\begin{array}{*{20}{l}}7&1&2\\9&2&1\end{array}} \right]\left[ {\begin{array}{*{20}{l}}3\\4\\5\end{array}} \right] + 2\left[ {\begin{array}{*{20}{l}}4\\2\end{array}} \right]$
A. $\left[ {\begin{array}{*{20}{l}}{43}\\{44}\end{array}} \right]$
В. $\left[ {\begin{array}{*{20}{l}}{43}\\{45}\end{array}} \right]$
C. $\left[ {\begin{array}{*{20}{l}}{45}\\{44}\end{array}} \right]$
D. $\left[ {\begin{array}{*{20}{l}}{44}\\{45}\end{array}} \right]$

Answer 1

Hint: If there are exactly as many columns in matrix A as there are rows in matrix B, the two matrices are said to be compatible. Accordingly, we can state that matrices A and B are compatible if A is a matrix of order $m \times n$ and B is a matrix of order $n \times p$ and in our case, we have a dimension of $3 \times 2$ and $3 \times 1$ to be multiplied and $2 \times 1$ with single element and both should be added in order to get the desired answer.

Formula Used: To multiply matrix, we use the formula
${C_{xy}} = {A_{x1}}{B_{y1}} + \ldots + {A_{xb}}{B_{by}} = \sum\limits_{k = 1}^b {{A_{xk}}} {B_{ky}}$

Complete step by step solution: We are provided the matrix in the question,
$\left[ {\begin{array}{*{20}{l}}7&1&2\\9&2&1\end{array}} \right]\left[ {\begin{array}{*{20}{l}}3\\4\\5\end{array}} \right] + 2\left[ {\begin{array}{*{20}{l}}4\\2\end{array}} \right]$
And are asked to determine the value it gives after solving the matrix.
Now, let us solve the one part of the given matrix expression.
That is,
$\left[ {\begin{array}{*{20}{l}}7&1&2\\9&2&1\end{array}} \right]\left[ {\begin{array}{*{20}{l}}3\\4\\5\end{array}} \right]$
For matrix multiplication, the elements in the first matrix's column and rows are multiplied by one another, and then the results are added.
$ = \left[ {\begin{array}{*{20}{l}}{(7) \cdot (3) + (1) \cdot (4) + (2) \cdot (5)}\\{(9) \cdot (3) + (2) \cdot (4) + (1) \cdot (5)}\end{array}} \right]$
Now, we have to solve the each terms in the determinant above, we get
$ = \left[ {\begin{array}{*{20}{l}}{21 + 4 + 10}\\{27 + 8 + 5}\end{array}} \right]$
Now, let us add the terms inside the determinant of each row, we get
$ = \left[ {\begin{array}{*{20}{l}}{35}\\{40}\end{array}} \right]$
Now, we have to solve the second part of the given matrix expression.
That is,
$2\left[ {\begin{array}{*{20}{l}}4\\2\end{array}} \right]$
To solve the above matrix, we have to multiply the element in the first matrix to each element in the second matrix, we get
$ = \left[ {\begin{array}{*{20}{l}}8\\4\end{array}} \right]$
Now, let us add both the results we get
$ = \left[ {\begin{array}{*{20}{l}}{35}\\{40}\end{array}} \right] + \left[ {\begin{array}{*{20}{l}}8\\4\end{array}} \right]$
Now, we have to simply add the corresponding entries from the two matrices and insert the result in the appropriate location of the resulting matrix to combine them.
Therefore, we get
$ = \left[ {\begin{array}{*{20}{l}}{(35) + (8)}\\{(40) + (4)}\end{array}} \right]$
On adding the each terms of the above matrix, we get
$ = \left[ {\begin{array}{*{20}{l}}{43}\\{44}\end{array}} \right]$
Therefore, the value of
$\left[ {\begin{array}{*{20}{l}}7&1&2\\9&2&1\end{array}} \right]\left[ {\begin{array}{*{20}{l}}3\\4\\5\end{array}} \right] + 2\left[ {\begin{array}{*{20}{l}}4\\2\end{array}} \right]$ is
$\left[ {\begin{array}{*{20}{l}}{43}\\{44}\end{array}} \right]$

Option ‘A’ is correct

Note: Here, students should keep in mind that for matrix multiplication, the first matrix's columns must match the rows of the second matrix in order to accomplish matrix multiplication. So, one should be careful while solving the matrix multiplication because the resulting matrix has the same number of columns and rows as the second matrix, and its number of columns matches the number of rows in the first matrix.