Video Transcript
Given that matrix 𝐴 equals 𝑖
cubed, 𝑖, zero, zero and matrix 𝐵 is also equal to 𝑖 cubed, 𝑖, zero, zero, and
𝑖 squared is equal to negative one, find 𝐴𝐵 if possible.
We recall that in order for matrix
multiplication to be defined, the number of columns in the first matrix must be
equal to the number of rows in the second matrix. In this question, the matrices 𝐴
and 𝐵 are both two-by-two square matrices. Therefore, 𝐴𝐵 is defined. And we’ll also have order two by
two. We need to multiply the matrix 𝑖
cubed, 𝑖, zero, zero by the same matrix 𝑖 cubed, 𝑖, zero, zero. When multiplying matrices, we need
to multiply the elements in each row of the first matrix by each column of the
second matrix. The top-left element is 𝑖 cubed
multiplied by 𝑖 cubed plus 𝑖 multiplied by zero. This is equal to 𝑖 to the sixth
power.
Next, we can multiply the first row
of the first matrix by the second column of the second matrix. 𝑖 cubed multiplied by 𝑖 plus 𝑖
multiplied by zero is equal to 𝑖 to the fourth power. We can then repeat this process for
the second row of our first matrix. As zero multiplied by anything is
equal to zero, both of these elements will equal zero. The matrix 𝐴𝐵 is equal to 𝑖 to
the sixth power, 𝑖 to the fourth power, zero, zero.
We are told in the question that 𝑖
squared equals negative one. 𝑖 to the fourth power is equal to
𝑖 squared squared. This means that 𝑖 to the fourth
power is equal to negative one squared, which is equal to one. 𝑖 to the sixth power is equal to
𝑖 squared all cubed. This is equal to negative one
cubed, which equals negative one. The matrix 𝐴𝐵 is therefore equal
to negative one, one, zero, zero.
