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

If 𝐴 is a matrix, which of the following is equal to (𝐴⁻¹)²? [A] 𝐴^(1/2) [B] 𝐴² [C] (𝐴⁻¹)^(1/2) [D] (𝐴²)⁻¹

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### 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.