### Video Transcript

If π΄ is a matrix, which of the
following is equal to π΄ inverse squared? π΄ to the half power, π΄ squared,
π΄ inverse to the half power, or π΄ squared inverse.

We can answer this question by
recalling the property of inverse matrices. That is, π΄ to the πth power
inverse equals π΄ inverse to the πth power, for π is a positive integer. So with this in mind, we can say
that π΄ inverse squared is equal to π΄ squared inverse. But letβs double-check this
relation just to be sure. We found that the inverse of π΄
squared is π΄ inverse squared. And we know if we take a matrix and
multiply it by its inverse, we should get the identity matrix. So letβs check this.

If we take the matrix π΄ squared
and multiply it by its inverse π΄ inverse squared, we should get the identity
matrix. We can just write this as π΄
multiplied by π΄ multiplied by π΄ inverse multiplied by π΄ inverse. And because of the associativity
property of matrix multiplication, we can write it in this way. We know π΄ multiplied by π΄ 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 π΄ multiplied by π΄
inverse, which gives us the identity matrix.