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Question: What is the product of \(\left[\begin{matrix} 1&2\\\3&4\end{matrix}\right]\) and \(\left[\begin{matr...

What is the product of [12\34]\left[\begin{matrix} 1&2\\\3&4\end{matrix}\right] and [567\8910]\left[\begin{matrix} 5&6&7\\\8&9&10\end{matrix}\right]?

Explanation

Solution

The matrix multiplication is a bit complex process because it is not done like the real numbers. We first need to ensure that the matrices we are multiplying have compatible orders or not. After that we multiply the first row by first column by element wise multiplication and summing them to form the element at the first column first row position. And we do it till we do not get the complete resultant matrix of the desired order.

Complete step-by-step solution:
To multiply two matrices we first check the basic required property of matching the order. Here, the first matrix has order 2×22\times 2 and the second matrix has order 2×32\times 3. And in order for the multiplication to be defined the number of columns of the first matrix must be equal to the number of rows of the second matrix. And in this case it is satisfied. Now, we use the formula below for matrix multiplication:
If A=[aij]A=[a_{ij}] is an m×nm\times n matrix and B=[bij]B=[b_{ij}] is an n×pn\times p matrix,
the product ABAB is an m×pm\times p matrix.
AB=[cij]AB=[c_{ij}]
where cij=ai1b1j+ai2b2j+...+ainbnjc_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+...+a_{in}b_{nj}
So we have:
A=[12\34]A=\left[\begin{matrix} 1&2\\\3&4\end{matrix}\right]
B=[567\8910]B=\left[\begin{matrix} 5&6&7\\\8&9&10\end{matrix}\right]
We have:
AB=[1×5+2×81×6+2×91×7+2×10\3×5+4×83×6+4×93×7+4×10]AB=\left[\begin{matrix} 1\times 5+2\times 8&1\times 6+2\times 9&1\times 7+2\times 10\\\3\times 5+4\times 8&3\times 6+4\times 9&3\times 7+4\times 10 \end{matrix}\right]
  ⟹  AB=[212427\475461]\implies AB=\left[\begin{matrix} 21&24&27\\\47&54&61 \end{matrix}\right]
Hence, we have found the product of the given matrices.

Note: Make sure that you add the terms before giving the resultant value in each position of the resultant matrix. Look for any calculation mistake that might occur while doing multiplication. Always check the order compatibility of the matrices before multiplying them. Also note that if we multiply the given matrices in the other order then matrix multiplication would not be possible.