Question Video: Finding the Inverse of the Transpose of a Matrix Mathematics

Given 𝐴⁻¹ = [1, 11 and 5, 17], find (𝐴^(𝑇))⁻¹.

02:54

Video Transcript

Given that 𝐴 inverse is the two-by-two matrix one, 11, five, 17, find the inverse of 𝐴 transpose.

In this question, we’re given the inverse of a matrix 𝐴. And we need to use this to determine the inverse of the transpose of matrix 𝐴. And there’s a lot of different ways we could go about this. For example, since we’re given the inverse of matrix 𝐴, we can invert this matrix to find 𝐴, find the transpose of this matrix, and then invert this again to find the inverse of the transpose of 𝐴. And although this method would work and give us the correct answer, there’s actually an easier method by using the properties of inverses of matrices.

We can, in fact, answer this question by recalling for any invertible matrix 𝐴 the inverse of the transpose of matrix 𝐴 is equal to the transpose of the inverse of 𝐴. And of course, we’re given in the question that 𝐴 is invertible. So we can use this to the question. The inverse of the transpose of 𝐴 is the transpose of the inverse of 𝐴. We can then substitute the two-by-two matrix we’re given for 𝐴 inverse into this expression. We get the transpose of the two-by-two matrix one, 11, five, 17.

Now all that’s left to do is evaluate the transpose of this matrix. And we recall to take the transpose of a matrix, we need to write the rows of this matrix as the column of our new matrix. Therefore, the first column of the transpose will be one, 11 and the second column will be five, 17. And this gives us our final answer. The inverse of the transpose of matrix 𝐴 is the two-by-two matrix one, five, 11, 17. And this is enough to answer our questions, so we could stop here. However, we could also discuss why this property holds true in the first place.

And to see why this property holds true, let’s prove it. Let’s start with an invertible matrix 𝑀. Since 𝑀 is invertible, the inverse of 𝑀 exists. In particular, 𝑀 multiplied by its inverse will be the identity matrix. We can take the transpose of both sides of this equation, giving us the following equation. We can simplify the left-hand side of this equation by noting We’re taking the transpose of the product of two matrices. And we recall this is the same as switching the order of our product and taking the transpose of each matrix individually. It’s the same as the transpose of 𝑀 inverse multiplied by the transpose of matrix 𝑀.

We can evaluate the right-hand side of this equation by noting we’re taking the transpose of the identity matrix. And remember, the identity matrix is a diagonal matrix. And this allows us to evaluate its transpose. It’s equal to itself, the identity matrix. And this allows us to see something interesting. We’re multiplying two matrices together and their product is the identity matrix, which means that these two matrices must be inverses of each other. And in particular, this means the transpose of the inverse of matrix 𝑀 is equal to the inverse of the transpose of matrix 𝑀. And therefore, we’ve proven this is true for any invertible matrix 𝑀. And we can apply this to our question to show that the inverse of the transpose of matrix 𝐴 is the two-by-two matrix one, five, 11, 17.

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