Video Transcript
Given the matrices 𝐴 and 𝐵, where
𝐴 equals one, negative two, three, zero, negative one, four, zero, zero, one and 𝐵
equals one, negative two, five, zero, negative one, four, zero, zero, one, find
𝐴𝐵. And the second part of the question
says, “Without doing any further calculations, find 𝐴 inverse.”
So the first thing we’re going to
do here is find the product 𝐴𝐵. Using the usual method for
multiplying three-by-three matrices together, we find that 𝐴𝐵 is one, zero, zero,
zero, one, zero, zero, zero, one. And we notice that this is actually
the three-by-three identity matrix. So what does this mean for our
matrices 𝐴 and 𝐵?
Well, the definition of the inverse
matrix is that it’s the 𝐴 inverse such that 𝐴 multiplied by 𝐴 inverse equals the
identity matrix. So the fact that we found the
product 𝐴𝐵 to be the identity matrix means that the matrix 𝐵 must be the inverse
of the matrix 𝐴.
The second part of the question
says, “Without doing any further calculations, find 𝐴 inverse.” Well, because when we find the
product 𝐴𝐵 we get the identity matrix, this means that the matrix 𝐵 is the
inverse of 𝐴. Therefore, 𝐴 inverse is the matrix
𝐵, which is one, negative two, five, zero, negative one, four, zero, zero, one.
