Matrix Multiplication

A matrix $A$ can be multiplied by a matrix $B$ if the number of columns in $A$ equals the number of rows in $B$. To find an element in the product matrix, multiply the elements of a row from the first matrix by the corresponding elements of a column from the second matrix, then add the products. $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \bullet \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$.

Find $AB$ if $A = \begin{bmatrix} 1 & 2 \\ 4 & 0 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 1 \\ 2 & 4 \end{bmatrix}$. The product is $AB = \begin{bmatrix} 1(5)+( 2)(2) & 1(1)+( 2)( 4) \\ 4(5)+0(2) & 4(1)+0( 4) \\ 2(5)+3(2) & 2(1)+3( 4) \end{bmatrix} = \begin{bmatrix} 1 & 9 \\ 20 & 4 \\ 16 & 10 \end{bmatrix}$. Find $XY$ if $X = \begin{bmatrix} 2 & 1 \\ 3 & 5 \end{bmatrix}$ and $Y = \begin{bmatrix} 0 & 4 \\ 1 & 2 \end{bmatrix}$. The product is $XY = \begin{bmatrix} 2(0)+( 1)(1) & 2(4)+( 1)( 2) \\ 3(0)+5(1) & 3(4)+5( 2) \end{bmatrix} = \begin{bmatrix} 1 & 10 \\ 5 & 2 \end{bmatrix}$.

Think of it as a 'row hugs column' dance! Each number in a row from the first matrix finds a partner in the corresponding column of the second matrix. They multiply, and all the pairs in that dance sum up to create one number in the final matrix. This repeats for every row and every column until the new matrix is complete.