Consider the matrices 𝐴 is equal
to one, two, negative seven and 𝐵 is equal to negative four, six, negative two. Find 𝐴𝐵 if possible.
We recall that matrix
multiplication is defined if the number of columns in the first matrix is equal to
the number of rows in the second matrix. For example, an 𝑚-by-𝑛 matrix
multiplied by an 𝑛-by-𝑘 matrix is defined. The resulting matrix will have 𝑚
rows and 𝑘 columns.
In this question, matrix 𝐴 has
order one by three, as there is one row and three columns. Matrix 𝐵 has order three by one,
as there are three rows and one column. This is known as a column
The number of columns in matrix 𝐴
is equal to the number of rows in matrix 𝐵. Therefore, the matrix 𝐴𝐵 will be
defined. The matrix 𝐴𝐵 will therefore have
one element equal to one multiplied by negative four plus two multiplied by six plus
negative seven multiplied by negative two. This simplifies to negative four
plus 12 plus 14, which in turn is equal to 22. 𝐴𝐵 is a matrix with one element,
the number 22.