# Question Video: Understanding Properties of Matrix Multiplication Applied to the Inverse of a Matrix Mathematics

If π΄ is a matrix, which of the following is equal to (π΄β»ΒΉ)Β²? [A] π΄^(1/2) [B] π΄Β² [C] (π΄β»ΒΉ)^(1/2) [D] (π΄Β²)β»ΒΉ

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### 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 πth power inverse equals π΄ inverse to the πth 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 letβs 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 letβs 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.