Question

Product of matrix $\left[ \begin{matrix} \text{a} & \text{b} & \text{c} \\\ \text{d} & \text{e} & \text{f} \\\ \text{h} & \text{i} & \text{j} \\\ \end{matrix} \right]\left[ \begin{matrix} \text{A} & \text{B} & \text{C} \\\ \text{D} & \text{E} & \text{F} \\\ \text{H} & \text{I} & \text{J} \\\ \end{matrix} \right]$ = ?

Answer 1

To solve this question we need to know the concept of multiplication of matrix. The first step in the multiplication of the matrix is to multiply the elements in the first row of the first matrix to the elements of the first column in the second matrix. Each element of a row of the first matrix is multiplied to the three columns of the second matrix in case of this question.

Complete step by step answer:
The question asks us to multiply two matrices of size $3\times 3$ and then find the result in matrix form. The product of two matrices can be computed by multiplying elements of the first row of the first matrix with the first column of the second matrix then adding all the product of elements. Continue this process until each row of the first matrix is multiplied with each column of the second matrix. Below is the way the matrix can be multiplied:
$\left[ \begin{matrix} \text{a} & \text{b} & \text{c} \\\ \text{d} & \text{e} & \text{f} \\\ \text{h} & \text{i} & \text{j} \\\ \end{matrix} \right]\left[ \begin{matrix} \text{A} & \text{B} & \text{C} \\\ \text{D} & \text{E} & \text{F} \\\ \text{H} & \text{I} & \text{J} \\\ \end{matrix} \right]=\left[ \begin{matrix} \text{a }\\!\\!\times\\!\\!\text{ A + b }\\!\\!\times\\!\\!\text{ D + c }\\!\\!\times\\!\\!\text{ H} & \text{a }\\!\\!\times\\!\\!\text{ B + b }\\!\\!\times\\!\\!\text{ E + c }\\!\\!\times\\!\\!\text{ I} & \text{a }\\!\\!\times\\!\\!\text{ C + b }\\!\\!\times\\!\\!\text{ F + c }\\!\\!\times\\!\\!\text{ J} \\\ \text{d }\\!\\!\times\\!\\!\text{ A + e }\\!\\!\times\\!\\!\text{ D + f }\\!\\!\times\\!\\!\text{ H} & \text{d }\\!\\!\times\\!\\!\text{ B + e }\\!\\!\times\\!\\!\text{ E + f }\\!\\!\times\\!\\!\text{ I} & \text{d }\\!\\!\times\\!\\!\text{ C + e }\\!\\!\times\\!\\!\text{ F + f }\\!\\!\times\\!\\!\text{ J} \\\ \text{h }\\!\\!\times\\!\\!\text{ A + i }\\!\\!\times\\!\\!\text{ D + j }\\!\\!\times\\!\\!\text{ H} & \text{h }\\!\\!\times\\!\\!\text{ B + i }\\!\\!\times\\!\\!\text{ E + j }\\!\\!\times\\!\\!\text{ I} & \text{h }\\!\\!\times\\!\\!\text{ C + i }\\!\\!\times\\!\\!\text{ F + j }\\!\\!\times\\!\\!\text{ J} \\\ \end{matrix} \right]$
Consider above example, first element in resulting matrix product $\left[ 0,0 \right]$ can be computed by multiplying first row of first matrix i.e. $\left( a,b,c \right)$ with first column of second matrix i.e. $\left( A,D,H \right)$ and finally sum all the product of elements i.e.$\left( a\times A \right)+\left( b\times D \right)+\left( c\times H \right)$ . Similarly, the second entry product $\left[ 0,1 \right]$ can be computed by multiplying the first row of the first matrix with the second column of the second matrix and sum all the products.

Note: Two matrices can be multiplied if and only if they satisfy the following condition: The number of columns present in the first matrix should be equal to the number of rows present in the second matrix. If the dimension of matrix A is p × q and matrix B is q × r, then the dimension of the resulting matrix will be p × r. The matrix address $\left[ 0,0 \right]$ means the element which is common in the first row and the first column, in case of the first matrix given in the question the value of $\left[ 0,0 \right]$ is $a$ .