Question Video: Properties of the Inverse and Transpose of a Matrix Mathematics

If 𝐴 is a matrix, which of the following is equal to (𝐴^(βˆ’1))^T? [A] (𝐴^T)^(βˆ’1) [B] 𝐴^(1/T) [C] (𝐴^(βˆ’1))^(1/T) [D] 𝐴^(βˆ’1) [E] 𝐴^T

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Video Transcript

If 𝐴 is a matrix, which of the following is equal to 𝐴 inverse transpose?

Recall that this notation means the transpose of a matrix. This just means we switch the rows with the columns. For example, if we have the matrix 𝑋 equals two, one, six, seven, 𝑋 transpose is equal to two, six, one, seven. To answer this question, we recall the property of the matrix inverse. That is, 𝐴 transposed inverse is equal to 𝐴 inverse transposed. And this gives us that our answer is the first option. 𝐴 inverse transposed is equal to 𝐴 transposed inverse. So what we’re saying is that if we take the matrix 𝐴, find its inverse, and then transpose it, this is exactly the same result we would get by taking the matrix 𝐴, transposing it, and then finding the inverse.

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