### Video Transcript

Find the value of π₯ which makes
the matrix π΄ equals negative one, five π₯ minus three, negative 43, negative eight
symmetric.

Remember, a square matrix π΄ is
symmetric if its transpose is equal to the original matrix. We see that π΄ is a square matrix;
itβs two by two. And so weβll be able to identify
the value of π₯ which makes the matrix symmetric by first finding its transpose. The transpose is found by switching
the rows and columns. And so since π΄ is a two-by-two
matrix, its transpose will also be two by two.

The first row of π΄ has elements
negative one and five π₯ 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 π₯
which make these two matrices equal. Now, of course, for two matrices to
be equal, we know their individual elements must also be equal. Letβs consider in particular the
element in the first row and second column. In our first matrix, that is five
π₯ minus three. And in the transpose, thatβs
negative 43. Since these are equal, we form and
solve an equation for π₯.

Itβs 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 π₯,
letβs add three to both sides. So five π₯ equals negative 40. Then we divide through by five, so
π₯ is equal to negative eight. The value of π₯ which makes the
matrix π΄ symmetric is negative eight.