### Video Transcript

Given the matrices π΄ and π΅, where
π΄ equals one, negative two, three, zero, negative one, four, zero, zero, one and π΅
equals one, negative two, five, zero, negative one, four, zero, zero, one, find
π΄π΅. And the second part of the question
says, βWithout doing any further calculations, find π΄ inverse.β

So the first thing weβre going to
do here is find the product π΄π΅. Using the usual method for
multiplying three-by-three matrices together, we find that π΄π΅ 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 π΄ and π΅?

Well, the definition of the inverse
matrix is that itβs the π΄ inverse such that π΄ multiplied by π΄ inverse equals the
identity matrix. So the fact that we found the
product π΄π΅ to be the identity matrix means that the matrix π΅ must be the inverse
of the matrix π΄.

The second part of the question
says, βWithout doing any further calculations, find π΄ inverse.β Well, because when we find the
product π΄π΅ we get the identity matrix, this means that the matrix π΅ is the
inverse of π΄. Therefore, π΄ inverse is the matrix
π΅, which is one, negative two, five, zero, negative one, four, zero, zero, one.