Question

Question: If \[U = \left[ {\begin{array}{*{20}{c}}2&{ - 3}&4\end{array}} \right]\], \[X = \left[ {\begin{array}{*{20}{c}}0&2&3\end{array}} \right]\], \[V = \left[ {\begin{array}{*{20}{c}}3\\2\\1\end{array}} \right]\]and\[Y = \left[ {\begin{array}{*{20}{c}}2\\2\\4\end{array}} \right]\], then what is the value of \[UV + XY\]?
A.\[\left[ {20} \right]\]
B.20
C. \[\left[ { - 20} \right]\]
D. -20

Answer 1

Hint: To solve the question we will first perform the multiplication operation on the matrices, and then perform the addition operation on the result we got from multiplying the matrices, then we will get the required result.
Formula Used:
We will use the multiplication of matrices with different dimensions, i.e.,
\[\left[ {\begin{array}{*{20}{c}}a&b&c\end{array}} \right]\left[ {\begin{array}{*{20}{c}}x\\y\\z\end{array}} \right] = \left[ {a \times x + b \times y + c \times z} \right]\]
Complete step by step solution:
Given matrices are \[U = \left[ {\begin{array}{*{20}{c}}2&{ - 3}&4\end{array}} \right]\], \[X = \left[ {\begin{array}{*{20}{c}}0&2&3\end{array}} \right]\], \[V = \left[ {\begin{array}{*{20}{c}}3\\2\\1\end{array}} \right]\]and\[Y = \left[ {\begin{array}{*{20}{c}}2\\2\\4\end{array}} \right]\]
Now we will multiply the matrices and then perform the addition operation on the matrices, we will get,
\[ \Rightarrow UV + XY = \left[ {\begin{array}{*{20}{c}}2&{ - 3}&4\end{array}} \right]\left[ {\begin{array}{*{20}{c}}3\\2\\1\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0&2&3\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2\\2\\4\end{array}} \right]\]
Now we will perform the multiplication operation first, we will get,
\[ \Rightarrow UV + XY = \left[ {2 \times 3 + \left( { - 3 \times 2} \right) + 4 \times 1} \right] + \left[ {0 \times 2 + 2 \times 2 + 3 \times 4} \right]\],
Now we will simplify by multiplying the terms, we will get,
\[ \Rightarrow UV + XY = \left[ {6 - 6 + 4} \right] + \left[ {0 + 4 + 12} \right]\]
Now we will further simplify by adding, then we will get,
\[ \Rightarrow UV + XY = \left[ 4 \right] + \left[ {16} \right]\]
Now we will perform the addition operation of the matrices, we will get,
\[ \Rightarrow UV + XY = \left[ {16 + 4} \right]\]
Now we will further simplify by adding, then we will get,
\[ \Rightarrow UV + XY = \left[ {20} \right]\]
The correct option is A.
Note: We know that to add two matrices, the dimensions of the two matrices should be same otherwise we cannot add the matrices. But in case of multiplication i.e., if we want to multiply any of the two matrices, the two matrices should have same inner dimensions, then the resulted matrix dimensions will be of the outer dimensions as its final dimension. It is important to know that the scalar multiplication of the matrix will not have any impact, as the original dimensions will be same as the resulted matrix. In other words the resulted matrix dimensions will be same as the original matrix dimension.