Question Video: Finding the Unknown Elements That Make a Given Matrix Symmetric Mathematics

Video Transcript

Find the value of 𝑥 which makes the matrix 𝐴 equals negative one, five 𝑥 minus three, negative 43, negative eight symmetric.

Remember, a square matrix 𝐴 is symmetric if its transpose is equal to the original matrix. We see that 𝐴 is a square matrix; it’s two by two. And so we’ll be able to identify the value of 𝑥 which makes the matrix symmetric by first finding its transpose. The transpose is found by switching the rows and columns. And so since 𝐴 is a two-by-two matrix, its transpose will also be two by two.

The first row of 𝐴 has elements negative one and five 𝑥 minus three. So this is the first column of the transpose. The second row has elements negative 43 and negative eight. So this is the second column. So we need to find the values of 𝑥 which make these two matrices equal. Now, of course, for two matrices to be equal, we know their individual elements must also be equal. Let’s consider in particular the element in the first row and second column. In our first matrix, that is five 𝑥 minus three. And in the transpose, that’s negative 43. Since these are equal, we form and solve an equation for 𝑥.

It’s worth also noting that had we instead equated the element in the second row and first column, we would have ended up with the same equation. To solve this equation for 𝑥, let’s add three to both sides. So five 𝑥 equals negative 40. Then we divide through by five, so 𝑥 is equal to negative eight. The value of 𝑥 which makes the matrix 𝐴 symmetric is negative eight.