# Question Video: Understanding the Properties of Inverse Matrices and the Identity Matrix Mathematics

Given the matrices 𝐴 and 𝐵, where 𝐴 = [1, −2, 3 and 0, −1, 4 and 0, 0, 1] and 𝐵 = [1, −2, 5 and 0, −1, 4 and 0, 0, 1], find 𝐴𝐵. Without doing any further calculations, find 𝐴⁻¹.

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### 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.