Video Transcript
If 𝐴 is a matrix, which of the
following is equal to 𝐴 inverse squared? 𝐴 to the half power, 𝐴 squared,
𝐴 inverse to the half power, or 𝐴 squared inverse.
We can answer this question by
recalling the property of inverse matrices. That is, 𝐴 to the 𝑛th power
inverse equals 𝐴 inverse to the 𝑛th power, for 𝑛 is a positive integer. So with this in mind, we can say
that 𝐴 inverse squared is equal to 𝐴 squared inverse. But let’s double-check this
relation just to be sure. We found that the inverse of 𝐴
squared is 𝐴 inverse squared. And we know if we take a matrix and
multiply it by its inverse, we should get the identity matrix. So let’s check this.
If we take the matrix 𝐴 squared
and multiply it by its inverse 𝐴 inverse squared, we should get the identity
matrix. We can just write this as 𝐴
multiplied by 𝐴 multiplied by 𝐴 inverse multiplied by 𝐴 inverse. And because of the associativity
property of matrix multiplication, we can write it in this way. We know 𝐴 multiplied by 𝐴 inverse
gives us the identity matrix. And we know multiplying any matrix
by the identity matrix just gives us the same matrix. So this is just 𝐴 multiplied by 𝐴
inverse, which gives us the identity matrix.
