Question Video: Finding Unknown Elements of a Symmetric Matrix Mathematics

Video Transcript

If 𝐴 — the matrix shown on screen — is a symmetric matrix, what are the values of 𝑥, 𝑦, and 𝑧?

A symmetric matrix is one that’s equal to its own transpose. 𝐴 is equal to 𝐴 transpose. The transpose of a matrix is found by swapping the rows and columns around. The first row of matrix 𝐴 becomes the first column of the matrix 𝐴 transpose, the second row of the matrix 𝐴 is the second column of the matrix 𝐴 transpose, the third row of the matrix 𝐴 is the third column of the matrix 𝐴 transpose.

As 𝐴 is a symmetric matrix and therefore 𝐴 is equal to 𝐴 transpose, this means that every element in 𝐴 is equal to its corresponding element in 𝐴 transpose. We can see this clearly for the elements on the diagonal which are just numbers. But it’s also true for the elements of the diagonal, which are mostly expressed in terms of the variables 𝑥, 𝑦, and 𝑧. By looking at corresponding elements between the two matrices, we can form some equations, which we can solve in order to find the values of 𝑥, 𝑦, and 𝑧.

For example, the element in the second row and third column of each matrix must be equal. Therefore, we have the equation 𝑧 minus six is equal to negative 11. Adding six to both sides of this equation gives the value of 𝑧. It’s equal to negative five. We would have got the same equation and hence the same value of 𝑧 if we’d instead equated the elements in the third row and second column of the two matrices.

Now, let’s look at what other elements we can equate in order to find the value of either 𝑥 or 𝑦. If we equate the elements in the first row and second column of the two matrices, then we have a relationship between 𝑦 and 𝑧. Negative two 𝑦 is equal to 𝑧 plus five. This is the same relationship that we would see if we equated the elements in the second row and first column.

We already know that 𝑧 is equal to negative five. So we can substitute this value of 𝑧 into the equation in order to solve for 𝑦. Negative two 𝑦 is equal to negative five plus five. Negative two 𝑦 is equal to zero. And dividing both sides of the equation by negative two, we see then that 𝑦 is equal to zero.

Next, we need to find an equation that we can use to find the value of 𝑥. If we equate the elements in the first row and third column of the two matrices, then we’ll have a relationship between 𝑦 and 𝑥. Negative 𝑦 minus three 𝑥 is equal to negative nine minus two 𝑥. Again, this same relationship could be achieved by equating the elements in the third row and first column. 𝑦 remember is equal to zero. So substituting this value into the equation gives zero minus three 𝑥 is equal to negative nine minus two 𝑥.

To solve this equation, we begin by adding two 𝑥 to each side. This gives negative 𝑥 is equal to negative nine. In order to find the value of 𝑥, we need to divide or multiply both sides of the equation by negative one. This gives 𝑥 is equal to nine.

The values of 𝑥, 𝑦, and 𝑧 found by equating corresponding elements of 𝐴 and 𝐴 transpose are 𝑥 is equal to nine, 𝑦 is equal to zero, 𝑧 is equal to negative five.